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@@ import{_ as n,c as e,j as t,a as s,a4 as Q,o as i}from"./chunks/framework.DGj8Ac plot!(result, "sqrt(u1^2 + v1^2)"; not_class="large")

Alternatively, we may visualise all underlying solutions, including complex ones,

julia
plot(result, "sqrt(u1^2 + v1^2)"; class="all")

2D parameters ​

',12)),t("p",null,[a[49]||(a[49]=s(`The parametrically driven oscillator boasts a stability diagram called "Arnold's tongues" delineating zones where the oscillator is stable from those where it is exponentially unstable (if the nonlinearity was absence). We can retrieve this diagram by calculating the steady states as a function of external detuning `)),t("mjx-container",Z,[(i(),e("svg",B,a[45]||(a[45]=[Q('',1)]))),a[46]||(a[46]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"δ"),t("mo",null,"="),t("msub",null,[t("mi",null,"ω"),t("mi",null,"L")]),t("mo",null,"−"),t("msub",null,[t("mi",null,"ω"),t("mn",null,"0")])])],-1))]),a[50]||(a[50]=s(" and the parametric drive strength ")),t("mjx-container",j,[(i(),e("svg",A,a[47]||(a[47]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),a[48]||(a[48]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"λ")])],-1))]),a[51]||(a[51]=s("."))]),t("p",null,[a[56]||(a[56]=s("To perform a 2D sweep over driving frequency ")),t("mjx-container",q,[(i(),e("svg",O,a[52]||(a[52]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[53]||(a[53]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[57]||(a[57]=s(" and parametric drive strength ")),t("mjx-container",R,[(i(),e("svg",S,a[54]||(a[54]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),a[55]||(a[55]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"λ")])],-1))]),a[58]||(a[58]=s(", we keep ")),a[59]||(a[59]=t("code",null,"fixed",-1)),a[60]||(a[60]=s(" from before but include 2 variables in ")),a[61]||(a[61]=t("code",null,"varied",-1))]),a[70]||(a[70]=Q(`
julia
fixed = (ω₀ => 1.0, γ => 1e-2, F => 1e-3, α => 1.0, η => 0.3)
 varied = (ω => range(0.8, 1.2, 50), λ => range(0.001, 0.6, 50))
 result_2D = get_steady_states(harmonic_eq, varied, fixed);

-Solving for 2500 parameters...  50%|██████████▏         |  ETA: 0:00:01\x1B[K
-  # parameters solved:  1260\x1B[K
-  # paths tracked:      6300\x1B[K
+Solving for 2500 parameters...  52%|██████████▍         |  ETA: 0:00:01\x1B[K
+  # parameters solved:  1301\x1B[K
+  # paths tracked:      6505\x1B[K
 \x1B[A
 \x1B[A
 
 
 \x1B[K\x1B[A
 \x1B[K\x1B[A
-Solving for 2500 parameters...  79%|███████████████▊    |  ETA: 0:00:00\x1B[K
-  # parameters solved:  1971\x1B[K
-  # paths tracked:      9855\x1B[K
+Solving for 2500 parameters...  81%|████████████████▎   |  ETA: 0:00:00\x1B[K
+  # parameters solved:  2031\x1B[K
+  # paths tracked:      10155\x1B[K
 \x1B[A
 \x1B[A
 
diff --git a/dev/assets/examples_parametron.md.BBoEju4P.lean.js b/dev/assets/examples_parametron.md.ApYdxv5d.lean.js
similarity index 99%
rename from dev/assets/examples_parametron.md.BBoEju4P.lean.js
rename to dev/assets/examples_parametron.md.ApYdxv5d.lean.js
index 59b2e200..33dd4509 100644
--- a/dev/assets/examples_parametron.md.BBoEju4P.lean.js
+++ b/dev/assets/examples_parametron.md.ApYdxv5d.lean.js
@@ -36,18 +36,18 @@ import{_ as n,c as e,j as t,a as s,a4 as Q,o as i}from"./chunks/framework.DGj8Ac
 plot!(result, "sqrt(u1^2 + v1^2)"; not_class="large")

Alternatively, we may visualise all underlying solutions, including complex ones,

julia
plot(result, "sqrt(u1^2 + v1^2)"; class="all")

2D parameters ​

',12)),t("p",null,[a[49]||(a[49]=s(`The parametrically driven oscillator boasts a stability diagram called "Arnold's tongues" delineating zones where the oscillator is stable from those where it is exponentially unstable (if the nonlinearity was absence). We can retrieve this diagram by calculating the steady states as a function of external detuning `)),t("mjx-container",Z,[(i(),e("svg",B,a[45]||(a[45]=[Q('',1)]))),a[46]||(a[46]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"δ"),t("mo",null,"="),t("msub",null,[t("mi",null,"ω"),t("mi",null,"L")]),t("mo",null,"−"),t("msub",null,[t("mi",null,"ω"),t("mn",null,"0")])])],-1))]),a[50]||(a[50]=s(" and the parametric drive strength ")),t("mjx-container",j,[(i(),e("svg",A,a[47]||(a[47]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),a[48]||(a[48]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"λ")])],-1))]),a[51]||(a[51]=s("."))]),t("p",null,[a[56]||(a[56]=s("To perform a 2D sweep over driving frequency ")),t("mjx-container",q,[(i(),e("svg",O,a[52]||(a[52]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[53]||(a[53]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[57]||(a[57]=s(" and parametric drive strength ")),t("mjx-container",R,[(i(),e("svg",S,a[54]||(a[54]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),a[55]||(a[55]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"λ")])],-1))]),a[58]||(a[58]=s(", we keep ")),a[59]||(a[59]=t("code",null,"fixed",-1)),a[60]||(a[60]=s(" from before but include 2 variables in ")),a[61]||(a[61]=t("code",null,"varied",-1))]),a[70]||(a[70]=Q(`
julia
fixed = (ω₀ => 1.0, γ => 1e-2, F => 1e-3, α => 1.0, η => 0.3)
 varied = (ω => range(0.8, 1.2, 50), λ => range(0.001, 0.6, 50))
 result_2D = get_steady_states(harmonic_eq, varied, fixed);

-Solving for 2500 parameters...  50%|██████████▏         |  ETA: 0:00:01\x1B[K
-  # parameters solved:  1260\x1B[K
-  # paths tracked:      6300\x1B[K
+Solving for 2500 parameters...  52%|██████████▍         |  ETA: 0:00:01\x1B[K
+  # parameters solved:  1301\x1B[K
+  # paths tracked:      6505\x1B[K
 \x1B[A
 \x1B[A
 
 
 \x1B[K\x1B[A
 \x1B[K\x1B[A
-Solving for 2500 parameters...  79%|███████████████▊    |  ETA: 0:00:00\x1B[K
-  # parameters solved:  1971\x1B[K
-  # paths tracked:      9855\x1B[K
+Solving for 2500 parameters...  81%|████████████████▎   |  ETA: 0:00:00\x1B[K
+  # parameters solved:  2031\x1B[K
+  # paths tracked:      10155\x1B[K
 \x1B[A
 \x1B[A
 
diff --git a/dev/assets/manual_Krylov-Bogoliubov_method.md.OSi_uHs9.js b/dev/assets/manual_Krylov-Bogoliubov_method.md.C4zEJfOC.js
similarity index 99%
rename from dev/assets/manual_Krylov-Bogoliubov_method.md.OSi_uHs9.js
rename to dev/assets/manual_Krylov-Bogoliubov_method.md.C4zEJfOC.js
index 9c22905e..fa7427d5 100644
--- a/dev/assets/manual_Krylov-Bogoliubov_method.md.OSi_uHs9.js
+++ b/dev/assets/manual_Krylov-Bogoliubov_method.md.C4zEJfOC.js
@@ -26,4 +26,4 @@ import{_ as h,c as t,j as i,a,a4 as n,G as k,B as p,o as e}from"./chunks/framewo
 
 ((1//2)*(ω^2)*v1(T) - (1//2)*(ω0^2)*v1(T)) / ω ~ Differential(T)(u1(T))
 
-((1//2)*(ω0^2)*u1(T) - (1//2)*F - (1//2)*(ω^2)*u1(T)) / ω ~ Differential(T)(v1(T))

source

`,7))]),s[13]||(s[13]=i("p",null,[a("For further information and a detailed understanding of this method, refer to "),i("a",{href:"https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogoliubov_averaging_method",target:"_blank",rel:"noreferrer"},"Krylov-Bogoliubov averaging method on Wikipedia"),a(".")],-1))])}const f=h(r,[["render",E]]);export{C as __pageData,f as default}; +((1//2)*(ω0^2)*u1(T) - (1//2)*F - (1//2)*(ω^2)*u1(T)) / ω ~ Differential(T)(v1(T))

source

`,7))]),s[13]||(s[13]=i("p",null,[a("For further information and a detailed understanding of this method, refer to "),i("a",{href:"https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogoliubov_averaging_method",target:"_blank",rel:"noreferrer"},"Krylov-Bogoliubov averaging method on Wikipedia"),a(".")],-1))])}const f=h(r,[["render",E]]);export{C as __pageData,f as default}; diff --git a/dev/assets/manual_Krylov-Bogoliubov_method.md.OSi_uHs9.lean.js b/dev/assets/manual_Krylov-Bogoliubov_method.md.C4zEJfOC.lean.js similarity index 99% rename from dev/assets/manual_Krylov-Bogoliubov_method.md.OSi_uHs9.lean.js rename to dev/assets/manual_Krylov-Bogoliubov_method.md.C4zEJfOC.lean.js index 9c22905e..fa7427d5 100644 --- a/dev/assets/manual_Krylov-Bogoliubov_method.md.OSi_uHs9.lean.js +++ b/dev/assets/manual_Krylov-Bogoliubov_method.md.C4zEJfOC.lean.js @@ -26,4 +26,4 @@ import{_ as h,c as t,j as i,a,a4 as n,G as k,B as p,o as e}from"./chunks/framewo ((1//2)*(ω^2)*v1(T) - (1//2)*(ω0^2)*v1(T)) / ω ~ Differential(T)(u1(T)) -((1//2)*(ω0^2)*u1(T) - (1//2)*F - (1//2)*(ω^2)*u1(T)) / ω ~ Differential(T)(v1(T))

source

`,7))]),s[13]||(s[13]=i("p",null,[a("For further information and a detailed understanding of this method, refer to "),i("a",{href:"https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogoliubov_averaging_method",target:"_blank",rel:"noreferrer"},"Krylov-Bogoliubov averaging method on Wikipedia"),a(".")],-1))])}const f=h(r,[["render",E]]);export{C as __pageData,f as default}; +((1//2)*(ω0^2)*u1(T) - (1//2)*F - (1//2)*(ω^2)*u1(T)) / ω ~ Differential(T)(v1(T))

source

`,7))]),s[13]||(s[13]=i("p",null,[a("For further information and a detailed understanding of this method, refer to "),i("a",{href:"https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogoliubov_averaging_method",target:"_blank",rel:"noreferrer"},"Krylov-Bogoliubov averaging method on Wikipedia"),a(".")],-1))])}const f=h(r,[["render",E]]);export{C as __pageData,f as default}; diff --git a/dev/assets/manual_entering_eom.md.DampzHDw.js b/dev/assets/manual_entering_eom.md.Ybl7g0Mg.js similarity index 98% rename from dev/assets/manual_entering_eom.md.DampzHDw.js rename to dev/assets/manual_entering_eom.md.Ybl7g0Mg.js index eb3c4126..1d3f0189 100644 --- a/dev/assets/manual_entering_eom.md.DampzHDw.js +++ b/dev/assets/manual_entering_eom.md.Ybl7g0Mg.js @@ -5,7 +5,7 @@ import{_ as e,c as h,j as s,a,G as t,a4 as l,B as k,o as p}from"./chunks/framewo julia> DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x); # two coupled oscillators, one of them driven -julia> DifferentialEquation([d(x,t,2) + ω0^2 * x - k*y, d(y,t,2) + ω0^2 * y - k*x] .~ [F * cos(ω*t), 0], [x,y]);

source

`,7))]),s("details",E,[s("summary",null,[i[3]||(i[3]=s("a",{id:"HarmonicBalance.add_harmonic!",href:"#HarmonicBalance.add_harmonic!"},[s("span",{class:"jlbinding"},"HarmonicBalance.add_harmonic!")],-1)),i[4]||(i[4]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[5]||(i[5]=l(`
julia
add_harmonic!(diff_eom::DifferentialEquation, var::Num, ω)

Add the harmonic ω to the harmonic ansatz used to expand the variable var in diff_eom.

Example

define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω

julia
julia> @variables t, x(t), y(t), ω0, ω, F, k;
+julia> DifferentialEquation([d(x,t,2) + ω0^2 * x - k*y, d(y,t,2) + ω0^2 * y - k*x] .~ [F * cos(ω*t), 0], [x,y]);

source

`,7))]),s("details",E,[s("summary",null,[i[3]||(i[3]=s("a",{id:"HarmonicBalance.add_harmonic!",href:"#HarmonicBalance.add_harmonic!"},[s("span",{class:"jlbinding"},"HarmonicBalance.add_harmonic!")],-1)),i[4]||(i[4]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[5]||(i[5]=l(`
julia
add_harmonic!(diff_eom::DifferentialEquation, var::Num, ω)

Add the harmonic ω to the harmonic ansatz used to expand the variable var in diff_eom.

Example

define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω

julia
julia> @variables t, x(t), y(t), ω0, ω, F, k;
 julia> diff_eq = DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x);
 julia> add_harmonic!(diff_eq, x, ω) # expand x using ω
 
@@ -13,6 +13,6 @@ import{_ as e,c as h,j as s,a,G as t,a4 as l,B as k,o as p}from"./chunks/framewo
 Variables:       x(t)
 Harmonic ansatz: x(t) => ω;
 
-(ω0^2)*x(t) + Differential(t)(Differential(t)(x(t))) ~ F*cos(t*ω)

source

`,6))]),s("details",o,[s("summary",null,[i[6]||(i[6]=s("a",{id:"Symbolics.get_variables-Tuple{DifferentialEquation}",href:"#Symbolics.get_variables-Tuple{DifferentialEquation}"},[s("span",{class:"jlbinding"},"Symbolics.get_variables")],-1)),i[7]||(i[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[8]||(i[8]=l('
julia
get_variables(diff_eom::DifferentialEquation) -> Vector{Num}

Return the dependent variables of diff_eom.

source

',3))]),s("details",g,[s("summary",null,[i[9]||(i[9]=s("a",{id:"HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}",href:"#HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_independent_variables")],-1)),i[10]||(i[10]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[11]||(i[11]=l(`
julia
get_independent_variables(
+(ω0^2)*x(t) + Differential(t)(Differential(t)(x(t))) ~ F*cos(t*ω)

source

`,6))]),s("details",o,[s("summary",null,[i[6]||(i[6]=s("a",{id:"Symbolics.get_variables-Tuple{DifferentialEquation}",href:"#Symbolics.get_variables-Tuple{DifferentialEquation}"},[s("span",{class:"jlbinding"},"Symbolics.get_variables")],-1)),i[7]||(i[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[8]||(i[8]=l('
julia
get_variables(diff_eom::DifferentialEquation) -> Vector{Num}

Return the dependent variables of diff_eom.

source

',3))]),s("details",g,[s("summary",null,[i[9]||(i[9]=s("a",{id:"HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}",href:"#HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_independent_variables")],-1)),i[10]||(i[10]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[11]||(i[11]=l(`
julia
get_independent_variables(
     diff_eom::DifferentialEquation
-) -> Any

Return the independent dependent variables of diff_eom.

source

`,3))])])}const D=e(r,[["render",y]]);export{b as __pageData,D as default}; +) -> Any

Return the independent dependent variables of diff_eom.

source

`,3))])])}const D=e(r,[["render",y]]);export{b as __pageData,D as default}; diff --git a/dev/assets/manual_entering_eom.md.DampzHDw.lean.js b/dev/assets/manual_entering_eom.md.Ybl7g0Mg.lean.js similarity index 98% rename from dev/assets/manual_entering_eom.md.DampzHDw.lean.js rename to dev/assets/manual_entering_eom.md.Ybl7g0Mg.lean.js index eb3c4126..1d3f0189 100644 --- a/dev/assets/manual_entering_eom.md.DampzHDw.lean.js +++ b/dev/assets/manual_entering_eom.md.Ybl7g0Mg.lean.js @@ -5,7 +5,7 @@ import{_ as e,c as h,j as s,a,G as t,a4 as l,B as k,o as p}from"./chunks/framewo julia> DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x); # two coupled oscillators, one of them driven -julia> DifferentialEquation([d(x,t,2) + ω0^2 * x - k*y, d(y,t,2) + ω0^2 * y - k*x] .~ [F * cos(ω*t), 0], [x,y]);

source

`,7))]),s("details",E,[s("summary",null,[i[3]||(i[3]=s("a",{id:"HarmonicBalance.add_harmonic!",href:"#HarmonicBalance.add_harmonic!"},[s("span",{class:"jlbinding"},"HarmonicBalance.add_harmonic!")],-1)),i[4]||(i[4]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[5]||(i[5]=l(`
julia
add_harmonic!(diff_eom::DifferentialEquation, var::Num, ω)

Add the harmonic ω to the harmonic ansatz used to expand the variable var in diff_eom.

Example

define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω

julia
julia> @variables t, x(t), y(t), ω0, ω, F, k;
+julia> DifferentialEquation([d(x,t,2) + ω0^2 * x - k*y, d(y,t,2) + ω0^2 * y - k*x] .~ [F * cos(ω*t), 0], [x,y]);

source

`,7))]),s("details",E,[s("summary",null,[i[3]||(i[3]=s("a",{id:"HarmonicBalance.add_harmonic!",href:"#HarmonicBalance.add_harmonic!"},[s("span",{class:"jlbinding"},"HarmonicBalance.add_harmonic!")],-1)),i[4]||(i[4]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[5]||(i[5]=l(`
julia
add_harmonic!(diff_eom::DifferentialEquation, var::Num, ω)

Add the harmonic ω to the harmonic ansatz used to expand the variable var in diff_eom.

Example

define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω

julia
julia> @variables t, x(t), y(t), ω0, ω, F, k;
 julia> diff_eq = DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x);
 julia> add_harmonic!(diff_eq, x, ω) # expand x using ω
 
@@ -13,6 +13,6 @@ import{_ as e,c as h,j as s,a,G as t,a4 as l,B as k,o as p}from"./chunks/framewo
 Variables:       x(t)
 Harmonic ansatz: x(t) => ω;
 
-(ω0^2)*x(t) + Differential(t)(Differential(t)(x(t))) ~ F*cos(t*ω)

source

`,6))]),s("details",o,[s("summary",null,[i[6]||(i[6]=s("a",{id:"Symbolics.get_variables-Tuple{DifferentialEquation}",href:"#Symbolics.get_variables-Tuple{DifferentialEquation}"},[s("span",{class:"jlbinding"},"Symbolics.get_variables")],-1)),i[7]||(i[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[8]||(i[8]=l('
julia
get_variables(diff_eom::DifferentialEquation) -> Vector{Num}

Return the dependent variables of diff_eom.

source

',3))]),s("details",g,[s("summary",null,[i[9]||(i[9]=s("a",{id:"HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}",href:"#HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_independent_variables")],-1)),i[10]||(i[10]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[11]||(i[11]=l(`
julia
get_independent_variables(
+(ω0^2)*x(t) + Differential(t)(Differential(t)(x(t))) ~ F*cos(t*ω)

source

`,6))]),s("details",o,[s("summary",null,[i[6]||(i[6]=s("a",{id:"Symbolics.get_variables-Tuple{DifferentialEquation}",href:"#Symbolics.get_variables-Tuple{DifferentialEquation}"},[s("span",{class:"jlbinding"},"Symbolics.get_variables")],-1)),i[7]||(i[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[8]||(i[8]=l('
julia
get_variables(diff_eom::DifferentialEquation) -> Vector{Num}

Return the dependent variables of diff_eom.

source

',3))]),s("details",g,[s("summary",null,[i[9]||(i[9]=s("a",{id:"HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}",href:"#HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_independent_variables")],-1)),i[10]||(i[10]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[11]||(i[11]=l(`
julia
get_independent_variables(
     diff_eom::DifferentialEquation
-) -> Any

Return the independent dependent variables of diff_eom.

source

`,3))])])}const D=e(r,[["render",y]]);export{b as __pageData,D as default}; +) -> Any

Return the independent dependent variables of diff_eom.

source

`,3))])])}const D=e(r,[["render",y]]);export{b as __pageData,D as default}; diff --git a/dev/assets/manual_extracting_harmonics.md.CIOZjrwR.lean.js b/dev/assets/manual_extracting_harmonics.md.hd0gVEbJ.js similarity index 99% rename from dev/assets/manual_extracting_harmonics.md.CIOZjrwR.lean.js rename to dev/assets/manual_extracting_harmonics.md.hd0gVEbJ.js index 36f27dc1..5406275b 100644 --- a/dev/assets/manual_extracting_harmonics.md.CIOZjrwR.lean.js +++ b/dev/assets/manual_extracting_harmonics.md.hd0gVEbJ.js @@ -20,13 +20,13 @@ import{_ as T,c as e,a4 as t,j as s,a as i,G as o,B as Q,o as n}from"./chunks/fr (ω0^2)*u1(T) + (2//1)*ω*Differential(T)(v1(T)) - (ω^2)*u1(T) ~ F -(ω0^2)*v1(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) ~ 0

source

`,7))]),s("details",d,[s("summary",null,[a[3]||(a[3]=s("a",{id:"HarmonicBalance.harmonic_ansatz",href:"#HarmonicBalance.harmonic_ansatz"},[s("span",{class:"jlbinding"},"HarmonicBalance.harmonic_ansatz")],-1)),a[4]||(a[4]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[5]||(a[5]=t('
julia
harmonic_ansatz(eom::DifferentialEquation, time::Num; coordinates="Cartesian")

Expand each variable of diff_eom using the harmonics assigned to it with time as the time variable. For each harmonic of each variable, instance(s) of HarmonicVariable are automatically created and named.

source

',3))]),s("details",p,[s("summary",null,[a[6]||(a[6]=s("a",{id:"HarmonicBalance.slow_flow",href:"#HarmonicBalance.slow_flow"},[s("span",{class:"jlbinding"},"HarmonicBalance.slow_flow")],-1)),a[7]||(a[7]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[8]||(a[8]=t('
julia
slow_flow(eom::HarmonicEquation; fast_time::Num, slow_time::Num, degree=2)

Removes all derivatives w.r.t fast_time (and their products) in eom of power degree. In the remaining derivatives, fast_time is replaced by slow_time.

source

',3))]),s("details",k,[s("summary",null,[a[9]||(a[9]=s("a",{id:"HarmonicBalance.fourier_transform",href:"#HarmonicBalance.fourier_transform"},[s("span",{class:"jlbinding"},"HarmonicBalance.fourier_transform")],-1)),a[10]||(a[10]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[11]||(a[11]=t(`
julia
fourier_transform(
+(ω0^2)*v1(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) ~ 0

source

`,7))]),s("details",d,[s("summary",null,[a[3]||(a[3]=s("a",{id:"HarmonicBalance.harmonic_ansatz",href:"#HarmonicBalance.harmonic_ansatz"},[s("span",{class:"jlbinding"},"HarmonicBalance.harmonic_ansatz")],-1)),a[4]||(a[4]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[5]||(a[5]=t('
julia
harmonic_ansatz(eom::DifferentialEquation, time::Num; coordinates="Cartesian")

Expand each variable of diff_eom using the harmonics assigned to it with time as the time variable. For each harmonic of each variable, instance(s) of HarmonicVariable are automatically created and named.

source

',3))]),s("details",p,[s("summary",null,[a[6]||(a[6]=s("a",{id:"HarmonicBalance.slow_flow",href:"#HarmonicBalance.slow_flow"},[s("span",{class:"jlbinding"},"HarmonicBalance.slow_flow")],-1)),a[7]||(a[7]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[8]||(a[8]=t('
julia
slow_flow(eom::HarmonicEquation; fast_time::Num, slow_time::Num, degree=2)

Removes all derivatives w.r.t fast_time (and their products) in eom of power degree. In the remaining derivatives, fast_time is replaced by slow_time.

source

',3))]),s("details",k,[s("summary",null,[a[9]||(a[9]=s("a",{id:"HarmonicBalance.fourier_transform",href:"#HarmonicBalance.fourier_transform"},[s("span",{class:"jlbinding"},"HarmonicBalance.fourier_transform")],-1)),a[10]||(a[10]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[11]||(a[11]=t(`
julia
fourier_transform(
     eom::HarmonicEquation,
     time::Num
-) -> HarmonicEquation

Extract the Fourier components of eom corresponding to the harmonics specified in eom.variables. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated time does not appear in the resulting equations anymore.

Underlying assumption: all time-dependences are harmonic.

source

`,4))]),s("details",m,[s("summary",null,[a[12]||(a[12]=s("a",{id:"HarmonicBalance.ExprUtils.drop_powers",href:"#HarmonicBalance.ExprUtils.drop_powers"},[s("span",{class:"jlbinding"},"HarmonicBalance.ExprUtils.drop_powers")],-1)),a[13]||(a[13]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[14]||(a[14]=t(`
julia
drop_powers(expr, vars, deg)

Remove parts of expr where the combined power of vars is => deg.

Example

julia
julia> @variables x,y;
+) -> HarmonicEquation

Extract the Fourier components of eom corresponding to the harmonics specified in eom.variables. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated time does not appear in the resulting equations anymore.

Underlying assumption: all time-dependences are harmonic.

source

`,4))]),s("details",m,[s("summary",null,[a[12]||(a[12]=s("a",{id:"HarmonicBalance.ExprUtils.drop_powers",href:"#HarmonicBalance.ExprUtils.drop_powers"},[s("span",{class:"jlbinding"},"HarmonicBalance.ExprUtils.drop_powers")],-1)),a[13]||(a[13]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[14]||(a[14]=t(`
julia
drop_powers(expr, vars, deg)

Remove parts of expr where the combined power of vars is => deg.

Example

julia
julia> @variables x,y;
 julia>drop_powers((x+y)^2, x, 2)
 y^2 + 2*x*y
 julia>drop_powers((x+y)^2, [x,y], 2)
 0
 julia>drop_powers((x+y)^2 + (x+y)^3, [x,y], 3)
-x^2 + y^2 + 2*x*y

source

`,5))]),a[57]||(a[57]=s("h2",{id:"HarmonicVariable-and-HarmonicEquation-types",tabindex:"-1"},[i("HarmonicVariable and HarmonicEquation types "),s("a",{class:"header-anchor",href:"#HarmonicVariable-and-HarmonicEquation-types","aria-label":'Permalink to "HarmonicVariable and HarmonicEquation types {#HarmonicVariable-and-HarmonicEquation-types}"'},"​")],-1)),s("p",null,[a[25]||(a[25]=i("The equations governing the harmonics are stored using the two following structs. When going from the original to the harmonic equations, the harmonic ansatz ")),s("mjx-container",g,[(n(),e("svg",c,a[15]||(a[15]=[t('',1)]))),a[16]||(a[16]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("munderover",null,[s("mo",{"data-mjx-texclass":"OP"},"∑"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"j"),s("mo",null,"="),s("mn",null,"1")]),s("mi",null,"M")]),s("msub",null,[s("mi",null,"u"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",null,"v"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[26]||(a[26]=i(" is used. Internally, each pair ")),s("mjx-container",u,[(n(),e("svg",y,a[17]||(a[17]=[t('',1)]))),a[18]||(a[18]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"u"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",null,","),s("msub",null,[s("mi",null,"v"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},")")])],-1))]),a[27]||(a[27]=i(" is stored as a ")),a[28]||(a[28]=s("code",null,"HarmonicVariable",-1)),a[29]||(a[29]=i(". This includes the identification of ")),s("mjx-container",E,[(n(),e("svg",f,a[19]||(a[19]=[t('',1)]))),a[20]||(a[20]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])])])],-1))]),a[30]||(a[30]=i(" and ")),s("mjx-container",x,[(n(),e("svg",w,a[21]||(a[21]=[t('',1)]))),a[22]||(a[22]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[31]||(a[31]=i(", which is needed to later reconstruct ")),s("mjx-container",H,[(n(),e("svg",b,a[23]||(a[23]=[t('',1)]))),a[24]||(a[24]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[32]||(a[32]=i("."))]),s("details",F,[s("summary",null,[a[33]||(a[33]=s("a",{id:"HarmonicBalance.HarmonicVariable",href:"#HarmonicBalance.HarmonicVariable"},[s("span",{class:"jlbinding"},"HarmonicBalance.HarmonicVariable")],-1)),a[34]||(a[34]=i()),o(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[35]||(a[35]=t('
julia
mutable struct HarmonicVariable

Holds a variable stored under symbol describing the harmonic ω of natural_variable.

Fields

  • symbol::Num: Symbol of the variable in the HarmonicBalance namespace.

  • name::String: Human-readable labels of the variable, used for plotting.

  • type::String: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)

  • ω::Num: The harmonic being described.

  • natural_variable::Num: The natural variable whose harmonic is being described.

source

',5))]),s("p",null,[a[44]||(a[44]=i("When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a ")),a[45]||(a[45]=s("code",null,"HarmonicEquation",-1)),a[46]||(a[46]=i(". 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julia
mutable struct HarmonicEquation

Holds a set of algebraic equations governing the harmonics of a DifferentialEquation.

Fields

  • equations::Vector{Equation}: A set of equations governing the harmonics.

  • variables::Vector{HarmonicVariable}: A set of variables describing the harmonics.

  • parameters::Vector{Num}: The parameters of the equation set.

  • natural_equation::DifferentialEquation: The natural equation (before the harmonic ansatz was used).

source

',5))])])}const G=T(r,[["render",A]]);export{S as __pageData,G as default}; +x^2 + y^2 + 2*x*y

source

`,5))]),a[57]||(a[57]=s("h2",{id:"HarmonicVariable-and-HarmonicEquation-types",tabindex:"-1"},[i("HarmonicVariable and HarmonicEquation types "),s("a",{class:"header-anchor",href:"#HarmonicVariable-and-HarmonicEquation-types","aria-label":'Permalink to "HarmonicVariable and HarmonicEquation types {#HarmonicVariable-and-HarmonicEquation-types}"'},"​")],-1)),s("p",null,[a[25]||(a[25]=i("The equations governing the harmonics are stored using the two following structs. When going from the original to the harmonic equations, the harmonic ansatz ")),s("mjx-container",g,[(n(),e("svg",c,a[15]||(a[15]=[t('',1)]))),a[16]||(a[16]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("munderover",null,[s("mo",{"data-mjx-texclass":"OP"},"∑"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"j"),s("mo",null,"="),s("mn",null,"1")]),s("mi",null,"M")]),s("msub",null,[s("mi",null,"u"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",null,"v"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[26]||(a[26]=i(" is used. Internally, each pair ")),s("mjx-container",u,[(n(),e("svg",y,a[17]||(a[17]=[t('',1)]))),a[18]||(a[18]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"u"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",null,","),s("msub",null,[s("mi",null,"v"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},")")])],-1))]),a[27]||(a[27]=i(" is stored as a ")),a[28]||(a[28]=s("code",null,"HarmonicVariable",-1)),a[29]||(a[29]=i(". This includes the identification of ")),s("mjx-container",E,[(n(),e("svg",f,a[19]||(a[19]=[t('',1)]))),a[20]||(a[20]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])])])],-1))]),a[30]||(a[30]=i(" and ")),s("mjx-container",x,[(n(),e("svg",w,a[21]||(a[21]=[t('',1)]))),a[22]||(a[22]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[31]||(a[31]=i(", which is needed to later reconstruct ")),s("mjx-container",H,[(n(),e("svg",b,a[23]||(a[23]=[t('',1)]))),a[24]||(a[24]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[32]||(a[32]=i("."))]),s("details",F,[s("summary",null,[a[33]||(a[33]=s("a",{id:"HarmonicBalance.HarmonicVariable",href:"#HarmonicBalance.HarmonicVariable"},[s("span",{class:"jlbinding"},"HarmonicBalance.HarmonicVariable")],-1)),a[34]||(a[34]=i()),o(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[35]||(a[35]=t('
julia
mutable struct HarmonicVariable

Holds a variable stored under symbol describing the harmonic ω of natural_variable.

Fields

  • symbol::Num: Symbol of the variable in the HarmonicBalance namespace.

  • name::String: Human-readable labels of the variable, used for plotting.

  • type::String: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)

  • ω::Num: The harmonic being described.

  • natural_variable::Num: The natural variable whose harmonic is being described.

source

',5))]),s("p",null,[a[44]||(a[44]=i("When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a ")),a[45]||(a[45]=s("code",null,"HarmonicEquation",-1)),a[46]||(a[46]=i(". For an initial equation of motion consisting of ")),s("mjx-container",v,[(n(),e("svg",L,a[36]||(a[36]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),a[37]||(a[37]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"M")])],-1))]),a[47]||(a[47]=i(" variables, each expanded in ")),s("mjx-container",C,[(n(),e("svg",D,a[38]||(a[38]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D441",d:"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z",style:{"stroke-width":"3"}})])])],-1)]))),a[39]||(a[39]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"N")])],-1))]),a[48]||(a[48]=i(" harmonics, the resulting ")),a[49]||(a[49]=s("code",null,"HarmonicEquation",-1)),a[50]||(a[50]=i(" holds ")),s("mjx-container",M,[(n(),e("svg",j,a[40]||(a[40]=[t('',1)]))),a[41]||(a[41]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"N"),s("mi",null,"M")])],-1))]),a[51]||(a[51]=i(" equations of ")),s("mjx-container",B,[(n(),e("svg",Z,a[42]||(a[42]=[t('',1)]))),a[43]||(a[43]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"N"),s("mi",null,"M")])],-1))]),a[52]||(a[52]=i(" variables. Each symbol not corresponding to a variable is identified as a parameter."))]),a[58]||(a[58]=s("p",null,[i("A "),s("code",null,"HarmonicEquation"),i(" can be either parsed into a steady-state "),s("code",null,"Problem"),i(" or solved using a dynamical ODE solver.")],-1)),s("details",V,[s("summary",null,[a[53]||(a[53]=s("a",{id:"HarmonicBalance.HarmonicEquation",href:"#HarmonicBalance.HarmonicEquation"},[s("span",{class:"jlbinding"},"HarmonicBalance.HarmonicEquation")],-1)),a[54]||(a[54]=i()),o(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[55]||(a[55]=t('
julia
mutable struct HarmonicEquation

Holds a set of algebraic equations governing the harmonics of a DifferentialEquation.

Fields

  • equations::Vector{Equation}: A set of equations governing the harmonics.

  • variables::Vector{HarmonicVariable}: A set of variables describing the harmonics.

  • parameters::Vector{Num}: The parameters of the equation set.

  • natural_equation::DifferentialEquation: The natural equation (before the harmonic ansatz was used).

source

',5))])])}const G=T(r,[["render",A]]);export{S as __pageData,G as default}; diff --git a/dev/assets/manual_extracting_harmonics.md.CIOZjrwR.js b/dev/assets/manual_extracting_harmonics.md.hd0gVEbJ.lean.js similarity index 99% rename from dev/assets/manual_extracting_harmonics.md.CIOZjrwR.js rename to dev/assets/manual_extracting_harmonics.md.hd0gVEbJ.lean.js index 36f27dc1..5406275b 100644 --- a/dev/assets/manual_extracting_harmonics.md.CIOZjrwR.js +++ b/dev/assets/manual_extracting_harmonics.md.hd0gVEbJ.lean.js @@ -20,13 +20,13 @@ import{_ as T,c as e,a4 as t,j as s,a as i,G as o,B as Q,o as n}from"./chunks/fr (ω0^2)*u1(T) + (2//1)*ω*Differential(T)(v1(T)) - (ω^2)*u1(T) ~ F -(ω0^2)*v1(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) ~ 0

source

`,7))]),s("details",d,[s("summary",null,[a[3]||(a[3]=s("a",{id:"HarmonicBalance.harmonic_ansatz",href:"#HarmonicBalance.harmonic_ansatz"},[s("span",{class:"jlbinding"},"HarmonicBalance.harmonic_ansatz")],-1)),a[4]||(a[4]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[5]||(a[5]=t('
julia
harmonic_ansatz(eom::DifferentialEquation, time::Num; coordinates="Cartesian")

Expand each variable of diff_eom using the harmonics assigned to it with time as the time variable. For each harmonic of each variable, instance(s) of HarmonicVariable are automatically created and named.

source

',3))]),s("details",p,[s("summary",null,[a[6]||(a[6]=s("a",{id:"HarmonicBalance.slow_flow",href:"#HarmonicBalance.slow_flow"},[s("span",{class:"jlbinding"},"HarmonicBalance.slow_flow")],-1)),a[7]||(a[7]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[8]||(a[8]=t('
julia
slow_flow(eom::HarmonicEquation; fast_time::Num, slow_time::Num, degree=2)

Removes all derivatives w.r.t fast_time (and their products) in eom of power degree. In the remaining derivatives, fast_time is replaced by slow_time.

source

',3))]),s("details",k,[s("summary",null,[a[9]||(a[9]=s("a",{id:"HarmonicBalance.fourier_transform",href:"#HarmonicBalance.fourier_transform"},[s("span",{class:"jlbinding"},"HarmonicBalance.fourier_transform")],-1)),a[10]||(a[10]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[11]||(a[11]=t(`
julia
fourier_transform(
+(ω0^2)*v1(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) ~ 0

source

`,7))]),s("details",d,[s("summary",null,[a[3]||(a[3]=s("a",{id:"HarmonicBalance.harmonic_ansatz",href:"#HarmonicBalance.harmonic_ansatz"},[s("span",{class:"jlbinding"},"HarmonicBalance.harmonic_ansatz")],-1)),a[4]||(a[4]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[5]||(a[5]=t('
julia
harmonic_ansatz(eom::DifferentialEquation, time::Num; coordinates="Cartesian")

Expand each variable of diff_eom using the harmonics assigned to it with time as the time variable. For each harmonic of each variable, instance(s) of HarmonicVariable are automatically created and named.

source

',3))]),s("details",p,[s("summary",null,[a[6]||(a[6]=s("a",{id:"HarmonicBalance.slow_flow",href:"#HarmonicBalance.slow_flow"},[s("span",{class:"jlbinding"},"HarmonicBalance.slow_flow")],-1)),a[7]||(a[7]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[8]||(a[8]=t('
julia
slow_flow(eom::HarmonicEquation; fast_time::Num, slow_time::Num, degree=2)

Removes all derivatives w.r.t fast_time (and their products) in eom of power degree. In the remaining derivatives, fast_time is replaced by slow_time.

source

',3))]),s("details",k,[s("summary",null,[a[9]||(a[9]=s("a",{id:"HarmonicBalance.fourier_transform",href:"#HarmonicBalance.fourier_transform"},[s("span",{class:"jlbinding"},"HarmonicBalance.fourier_transform")],-1)),a[10]||(a[10]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[11]||(a[11]=t(`
julia
fourier_transform(
     eom::HarmonicEquation,
     time::Num
-) -> HarmonicEquation

Extract the Fourier components of eom corresponding to the harmonics specified in eom.variables. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated time does not appear in the resulting equations anymore.

Underlying assumption: all time-dependences are harmonic.

source

`,4))]),s("details",m,[s("summary",null,[a[12]||(a[12]=s("a",{id:"HarmonicBalance.ExprUtils.drop_powers",href:"#HarmonicBalance.ExprUtils.drop_powers"},[s("span",{class:"jlbinding"},"HarmonicBalance.ExprUtils.drop_powers")],-1)),a[13]||(a[13]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[14]||(a[14]=t(`
julia
drop_powers(expr, vars, deg)

Remove parts of expr where the combined power of vars is => deg.

Example

julia
julia> @variables x,y;
+) -> HarmonicEquation

Extract the Fourier components of eom corresponding to the harmonics specified in eom.variables. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated time does not appear in the resulting equations anymore.

Underlying assumption: all time-dependences are harmonic.

source

`,4))]),s("details",m,[s("summary",null,[a[12]||(a[12]=s("a",{id:"HarmonicBalance.ExprUtils.drop_powers",href:"#HarmonicBalance.ExprUtils.drop_powers"},[s("span",{class:"jlbinding"},"HarmonicBalance.ExprUtils.drop_powers")],-1)),a[13]||(a[13]=i()),o(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[14]||(a[14]=t(`
julia
drop_powers(expr, vars, deg)

Remove parts of expr where the combined power of vars is => deg.

Example

julia
julia> @variables x,y;
 julia>drop_powers((x+y)^2, x, 2)
 y^2 + 2*x*y
 julia>drop_powers((x+y)^2, [x,y], 2)
 0
 julia>drop_powers((x+y)^2 + (x+y)^3, [x,y], 3)
-x^2 + y^2 + 2*x*y

source

`,5))]),a[57]||(a[57]=s("h2",{id:"HarmonicVariable-and-HarmonicEquation-types",tabindex:"-1"},[i("HarmonicVariable and HarmonicEquation types "),s("a",{class:"header-anchor",href:"#HarmonicVariable-and-HarmonicEquation-types","aria-label":'Permalink to "HarmonicVariable and HarmonicEquation types {#HarmonicVariable-and-HarmonicEquation-types}"'},"​")],-1)),s("p",null,[a[25]||(a[25]=i("The equations governing the harmonics are stored using the two following structs. When going from the original to the harmonic equations, the harmonic ansatz ")),s("mjx-container",g,[(n(),e("svg",c,a[15]||(a[15]=[t('',1)]))),a[16]||(a[16]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("munderover",null,[s("mo",{"data-mjx-texclass":"OP"},"∑"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"j"),s("mo",null,"="),s("mn",null,"1")]),s("mi",null,"M")]),s("msub",null,[s("mi",null,"u"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",null,"v"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[26]||(a[26]=i(" is used. Internally, each pair ")),s("mjx-container",u,[(n(),e("svg",y,a[17]||(a[17]=[t('',1)]))),a[18]||(a[18]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"u"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",null,","),s("msub",null,[s("mi",null,"v"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},")")])],-1))]),a[27]||(a[27]=i(" is stored as a ")),a[28]||(a[28]=s("code",null,"HarmonicVariable",-1)),a[29]||(a[29]=i(". This includes the identification of ")),s("mjx-container",E,[(n(),e("svg",f,a[19]||(a[19]=[t('',1)]))),a[20]||(a[20]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])])])],-1))]),a[30]||(a[30]=i(" and ")),s("mjx-container",x,[(n(),e("svg",w,a[21]||(a[21]=[t('',1)]))),a[22]||(a[22]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[31]||(a[31]=i(", which is needed to later reconstruct ")),s("mjx-container",H,[(n(),e("svg",b,a[23]||(a[23]=[t('',1)]))),a[24]||(a[24]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[32]||(a[32]=i("."))]),s("details",F,[s("summary",null,[a[33]||(a[33]=s("a",{id:"HarmonicBalance.HarmonicVariable",href:"#HarmonicBalance.HarmonicVariable"},[s("span",{class:"jlbinding"},"HarmonicBalance.HarmonicVariable")],-1)),a[34]||(a[34]=i()),o(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[35]||(a[35]=t('
julia
mutable struct HarmonicVariable

Holds a variable stored under symbol describing the harmonic ω of natural_variable.

Fields

  • symbol::Num: Symbol of the variable in the HarmonicBalance namespace.

  • name::String: Human-readable labels of the variable, used for plotting.

  • type::String: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)

  • ω::Num: The harmonic being described.

  • natural_variable::Num: The natural variable whose harmonic is being described.

source

',5))]),s("p",null,[a[44]||(a[44]=i("When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a ")),a[45]||(a[45]=s("code",null,"HarmonicEquation",-1)),a[46]||(a[46]=i(". For an initial equation of motion consisting of ")),s("mjx-container",v,[(n(),e("svg",L,a[36]||(a[36]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),a[37]||(a[37]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"M")])],-1))]),a[47]||(a[47]=i(" variables, each expanded in ")),s("mjx-container",C,[(n(),e("svg",D,a[38]||(a[38]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D441",d:"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z",style:{"stroke-width":"3"}})])])],-1)]))),a[39]||(a[39]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"N")])],-1))]),a[48]||(a[48]=i(" harmonics, the resulting ")),a[49]||(a[49]=s("code",null,"HarmonicEquation",-1)),a[50]||(a[50]=i(" holds ")),s("mjx-container",M,[(n(),e("svg",j,a[40]||(a[40]=[t('',1)]))),a[41]||(a[41]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"N"),s("mi",null,"M")])],-1))]),a[51]||(a[51]=i(" equations of ")),s("mjx-container",B,[(n(),e("svg",Z,a[42]||(a[42]=[t('',1)]))),a[43]||(a[43]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"N"),s("mi",null,"M")])],-1))]),a[52]||(a[52]=i(" variables. Each symbol not corresponding to a variable is identified as a parameter."))]),a[58]||(a[58]=s("p",null,[i("A "),s("code",null,"HarmonicEquation"),i(" can be either parsed into a steady-state "),s("code",null,"Problem"),i(" or solved using a dynamical ODE solver.")],-1)),s("details",V,[s("summary",null,[a[53]||(a[53]=s("a",{id:"HarmonicBalance.HarmonicEquation",href:"#HarmonicBalance.HarmonicEquation"},[s("span",{class:"jlbinding"},"HarmonicBalance.HarmonicEquation")],-1)),a[54]||(a[54]=i()),o(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[55]||(a[55]=t('
julia
mutable struct HarmonicEquation

Holds a set of algebraic equations governing the harmonics of a DifferentialEquation.

Fields

  • equations::Vector{Equation}: A set of equations governing the harmonics.

  • variables::Vector{HarmonicVariable}: A set of variables describing the harmonics.

  • parameters::Vector{Num}: The parameters of the equation set.

  • natural_equation::DifferentialEquation: The natural equation (before the harmonic ansatz was used).

source

',5))])])}const G=T(r,[["render",A]]);export{S as __pageData,G as default}; +x^2 + y^2 + 2*x*y

source

`,5))]),a[57]||(a[57]=s("h2",{id:"HarmonicVariable-and-HarmonicEquation-types",tabindex:"-1"},[i("HarmonicVariable and HarmonicEquation types "),s("a",{class:"header-anchor",href:"#HarmonicVariable-and-HarmonicEquation-types","aria-label":'Permalink to "HarmonicVariable and HarmonicEquation types {#HarmonicVariable-and-HarmonicEquation-types}"'},"​")],-1)),s("p",null,[a[25]||(a[25]=i("The equations governing the harmonics are stored using the two following structs. When going from the original to the harmonic equations, the harmonic ansatz ")),s("mjx-container",g,[(n(),e("svg",c,a[15]||(a[15]=[t('',1)]))),a[16]||(a[16]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("munderover",null,[s("mo",{"data-mjx-texclass":"OP"},"∑"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"j"),s("mo",null,"="),s("mn",null,"1")]),s("mi",null,"M")]),s("msub",null,[s("mi",null,"u"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",null,"v"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[26]||(a[26]=i(" is used. Internally, each pair ")),s("mjx-container",u,[(n(),e("svg",y,a[17]||(a[17]=[t('',1)]))),a[18]||(a[18]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"u"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",null,","),s("msub",null,[s("mi",null,"v"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])]),s("mo",{stretchy:"false"},")")])],-1))]),a[27]||(a[27]=i(" is stored as a ")),a[28]||(a[28]=s("code",null,"HarmonicVariable",-1)),a[29]||(a[29]=i(". This includes the identification of ")),s("mjx-container",E,[(n(),e("svg",f,a[19]||(a[19]=[t('',1)]))),a[20]||(a[20]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"i"),s("mo",null,","),s("mi",null,"j")])])])],-1))]),a[30]||(a[30]=i(" and ")),s("mjx-container",x,[(n(),e("svg",w,a[21]||(a[21]=[t('',1)]))),a[22]||(a[22]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[31]||(a[31]=i(", which is needed to later reconstruct ")),s("mjx-container",H,[(n(),e("svg",b,a[23]||(a[23]=[t('',1)]))),a[24]||(a[24]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[32]||(a[32]=i("."))]),s("details",F,[s("summary",null,[a[33]||(a[33]=s("a",{id:"HarmonicBalance.HarmonicVariable",href:"#HarmonicBalance.HarmonicVariable"},[s("span",{class:"jlbinding"},"HarmonicBalance.HarmonicVariable")],-1)),a[34]||(a[34]=i()),o(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[35]||(a[35]=t('
julia
mutable struct HarmonicVariable

Holds a variable stored under symbol describing the harmonic ω of natural_variable.

Fields

  • symbol::Num: Symbol of the variable in the HarmonicBalance namespace.

  • name::String: Human-readable labels of the variable, used for plotting.

  • type::String: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)

  • ω::Num: The harmonic being described.

  • natural_variable::Num: The natural variable whose harmonic is being described.

source

',5))]),s("p",null,[a[44]||(a[44]=i("When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a ")),a[45]||(a[45]=s("code",null,"HarmonicEquation",-1)),a[46]||(a[46]=i(". For an initial equation of motion consisting of ")),s("mjx-container",v,[(n(),e("svg",L,a[36]||(a[36]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),a[37]||(a[37]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"M")])],-1))]),a[47]||(a[47]=i(" variables, each expanded in ")),s("mjx-container",C,[(n(),e("svg",D,a[38]||(a[38]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D441",d:"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z",style:{"stroke-width":"3"}})])])],-1)]))),a[39]||(a[39]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"N")])],-1))]),a[48]||(a[48]=i(" harmonics, the resulting ")),a[49]||(a[49]=s("code",null,"HarmonicEquation",-1)),a[50]||(a[50]=i(" holds ")),s("mjx-container",M,[(n(),e("svg",j,a[40]||(a[40]=[t('',1)]))),a[41]||(a[41]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"N"),s("mi",null,"M")])],-1))]),a[51]||(a[51]=i(" equations of ")),s("mjx-container",B,[(n(),e("svg",Z,a[42]||(a[42]=[t('',1)]))),a[43]||(a[43]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"N"),s("mi",null,"M")])],-1))]),a[52]||(a[52]=i(" variables. Each symbol not corresponding to a variable is identified as a parameter."))]),a[58]||(a[58]=s("p",null,[i("A "),s("code",null,"HarmonicEquation"),i(" can be either parsed into a steady-state "),s("code",null,"Problem"),i(" or solved using a dynamical ODE solver.")],-1)),s("details",V,[s("summary",null,[a[53]||(a[53]=s("a",{id:"HarmonicBalance.HarmonicEquation",href:"#HarmonicBalance.HarmonicEquation"},[s("span",{class:"jlbinding"},"HarmonicBalance.HarmonicEquation")],-1)),a[54]||(a[54]=i()),o(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[55]||(a[55]=t('
julia
mutable struct HarmonicEquation

Holds a set of algebraic equations governing the harmonics of a DifferentialEquation.

Fields

  • equations::Vector{Equation}: A set of equations governing the harmonics.

  • variables::Vector{HarmonicVariable}: A set of variables describing the harmonics.

  • parameters::Vector{Num}: The parameters of the equation set.

  • natural_equation::DifferentialEquation: The natural equation (before the harmonic ansatz was used).

source

',5))])])}const G=T(r,[["render",A]]);export{S as __pageData,G as default}; diff --git a/dev/assets/manual_linear_response.md.CD1hVQFy.js b/dev/assets/manual_linear_response.md.CafqEdgB.js similarity index 95% rename from dev/assets/manual_linear_response.md.CD1hVQFy.js rename to dev/assets/manual_linear_response.md.CafqEdgB.js index 4b0231bb..b72c161c 100644 --- a/dev/assets/manual_linear_response.md.CD1hVQFy.js +++ b/dev/assets/manual_linear_response.md.CafqEdgB.js @@ -1,5 +1,5 @@ -import{_ as r,c as o,a4 as i,j as s,a,G as n,B as p,o as l}from"./chunks/framework.DGj8AcR1.js";const B=JSON.parse('{"title":"Linear response (WIP)","description":"","frontmatter":{},"headers":[],"relativePath":"manual/linear_response.md","filePath":"manual/linear_response.md"}'),d={name:"manual/linear_response.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},k={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.247ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2319 1000","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.278ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2333 1000","aria-hidden":"true"},f={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""};function H(v,e,j,w,L,F){const t=p("Badge");return l(),o("div",null,[e[31]||(e[31]=i('

Linear response (WIP) ​

This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.

The methodology used is explained in Jan Kosata phd thesis.

Stability ​

The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,

',5)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"HarmonicBalance.LinearResponse.get_Jacobian",href:"#HarmonicBalance.LinearResponse.get_Jacobian"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_Jacobian")],-1)),e[1]||(e[1]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=i('
julia
get_Jacobian(eom)

Obtain the symbolic Jacobian matrix of eom (either a HarmonicEquation or a DifferentialEquation). This is the linearised left-hand side of F(u) = du/dT.

source

Obtain a Jacobian from a DifferentialEquation by first converting it into a HarmonicEquation.

source

Get the Jacobian of a set of equations eqs with respect to the variables vars.

source

',7))]),e[32]||(e[32]=s("h2",{id:"Linear-response",tabindex:"-1"},[a("Linear response "),s("a",{class:"header-anchor",href:"#Linear-response","aria-label":'Permalink to "Linear response {#Linear-response}"'},"​")],-1)),e[33]||(e[33]=s("p",null,[a("The response to white noise can be shown with "),s("code",null,"plot_linear_response"),a(". Depending on the "),s("code",null,"order"),a(" argument, different methods are used.")],-1)),s("details",c,[s("summary",null,[e[3]||(e[3]=s("a",{id:"HarmonicBalance.LinearResponse.plot_linear_response",href:"#HarmonicBalance.LinearResponse.plot_linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_linear_response")],-1)),e[4]||(e[4]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[5]||(e[5]=i('
julia
plot_linear_response(res::Result, nat_var::Num; Ω_range, branch::Int, order=1, logscale=false, show_progress=true, kwargs...)

Plot the linear response to white noise of the variable nat_var for Result res on branch for input frequencies Ω_range. Slow-time derivatives up to order are kept in the process.

Any kwargs are fed to Plots' gr().

Solutions not belonging to the physical class are ignored.

source

',5))]),e[34]||(e[34]=s("h3",{id:"First-order",tabindex:"-1"},[a("First order "),s("a",{class:"header-anchor",href:"#First-order","aria-label":'Permalink to "First order {#First-order}"'},"​")],-1)),s("p",null,[e[12]||(e[12]=a("The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues ")),s("mjx-container",k,[(l(),o("svg",m,e[6]||(e[6]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),e[7]||(e[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"λ")])],-1))]),e[13]||(e[13]=a(" describes a Lorentzian peak in the response; ")),s("mjx-container",g,[(l(),o("svg",u,e[8]||(e[8]=[i('',1)]))),e[9]||(e[9]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Re"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[14]||(e[14]=a(" gives its center and ")),s("mjx-container",b,[(l(),o("svg",y,e[10]||(e[10]=[i('',1)]))),e[11]||(e[11]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Im"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[15]||(e[15]=a(" its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment."))]),e[35]||(e[35]=s("p",null,"The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).",-1)),e[36]||(e[36]=s("p",null,[a("Behind the scenes, the spectra are stored using the dedicated structs "),s("code",null,"Lorentzian"),a(" and "),s("code",null,"JacobianSpectrum"),a(".")],-1)),s("details",f,[s("summary",null,[e[16]||(e[16]=s("a",{id:"HarmonicBalance.LinearResponse.JacobianSpectrum",href:"#HarmonicBalance.LinearResponse.JacobianSpectrum"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.JacobianSpectrum")],-1)),e[17]||(e[17]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[18]||(e[18]=i('
julia
mutable struct JacobianSpectrum

Holds a set of Lorentzian objects belonging to a variable.

Fields

  • peaks::Vector{HarmonicBalance.LinearResponse.Lorentzian}

Constructor

julia
JacobianSpectrum(res::Result; index::Int, branch::Int)

source

',7))]),s("details",T,[s("summary",null,[e[19]||(e[19]=s("a",{id:"HarmonicBalance.LinearResponse.Lorentzian",href:"#HarmonicBalance.LinearResponse.Lorentzian"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.Lorentzian")],-1)),e[20]||(e[20]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[21]||(e[21]=i('
julia
struct Lorentzian

Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).

Fields

  • ω0::Float64

  • Γ::Float64

  • A::Float64

source

',5))]),e[37]||(e[37]=s("h3",{id:"Higher-orders",tabindex:"-1"},[a("Higher orders "),s("a",{class:"header-anchor",href:"#Higher-orders","aria-label":'Permalink to "Higher orders {#Higher-orders}"'},"​")],-1)),e[38]||(e[38]=s("p",null,[a("Setting "),s("code",null,"order > 1"),a(" increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.")],-1)),s("details",Q,[s("summary",null,[e[22]||(e[22]=s("a",{id:"HarmonicBalance.LinearResponse.ResponseMatrix",href:"#HarmonicBalance.LinearResponse.ResponseMatrix"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.ResponseMatrix")],-1)),e[23]||(e[23]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[24]||(e[24]=i('
julia
struct ResponseMatrix

Holds the compiled response matrix of a system.

Fields

  • matrix::Matrix{Function}: The response matrix (compiled).

  • symbols::Vector{Num}: Any symbolic variables in matrix to be substituted at evaluation.

  • variables::Vector{HarmonicVariable}: The frequencies of the harmonic variables underlying matrix. These are needed to transform the harmonic variables to the non-rotating frame.

source

',5))]),s("details",x,[s("summary",null,[e[25]||(e[25]=s("a",{id:"HarmonicBalance.LinearResponse.get_response",href:"#HarmonicBalance.LinearResponse.get_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response")],-1)),e[26]||(e[26]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[27]||(e[27]=i(`
julia
get_response(
+import{_ as r,c as o,a4 as i,j as s,a,G as n,B as p,o as l}from"./chunks/framework.DGj8AcR1.js";const B=JSON.parse('{"title":"Linear response (WIP)","description":"","frontmatter":{},"headers":[],"relativePath":"manual/linear_response.md","filePath":"manual/linear_response.md"}'),d={name:"manual/linear_response.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},k={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.247ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2319 1000","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.278ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2333 1000","aria-hidden":"true"},f={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""};function H(v,e,j,w,L,F){const t=p("Badge");return l(),o("div",null,[e[31]||(e[31]=i('
Linear response (WIP) ​
This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.
The methodology used is explained in Jan Kosata phd thesis.
Stability ​
The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,
',5)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"HarmonicBalance.LinearResponse.get_Jacobian",href:"#HarmonicBalance.LinearResponse.get_Jacobian"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_Jacobian")],-1)),e[1]||(e[1]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=i('
julia
get_Jacobian(eom)
Obtain the symbolic Jacobian matrix of eom (either a HarmonicEquation or a DifferentialEquation). This is the linearised left-hand side of F(u) = du/dT.
source
Obtain a Jacobian from a DifferentialEquation by first converting it into a HarmonicEquation.
source
Get the Jacobian of a set of equations eqs with respect to the variables vars.
source
',7))]),e[32]||(e[32]=s("h2",{id:"Linear-response",tabindex:"-1"},[a("Linear response "),s("a",{class:"header-anchor",href:"#Linear-response","aria-label":'Permalink to "Linear response {#Linear-response}"'},"​")],-1)),e[33]||(e[33]=s("p",null,[a("The response to white noise can be shown with "),s("code",null,"plot_linear_response"),a(". Depending on the "),s("code",null,"order"),a(" argument, different methods are used.")],-1)),s("details",c,[s("summary",null,[e[3]||(e[3]=s("a",{id:"HarmonicBalance.LinearResponse.plot_linear_response",href:"#HarmonicBalance.LinearResponse.plot_linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_linear_response")],-1)),e[4]||(e[4]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[5]||(e[5]=i('
julia
plot_linear_response(res::Result, nat_var::Num; Ω_range, branch::Int, order=1, logscale=false, show_progress=true, kwargs...)
Plot the linear response to white noise of the variable nat_var for Result res on branch for input frequencies Ω_range. Slow-time derivatives up to order are kept in the process.
Any kwargs are fed to Plots' gr().
Solutions not belonging to the physical class are ignored.
source
',5))]),e[34]||(e[34]=s("h3",{id:"First-order",tabindex:"-1"},[a("First order "),s("a",{class:"header-anchor",href:"#First-order","aria-label":'Permalink to "First order {#First-order}"'},"​")],-1)),s("p",null,[e[12]||(e[12]=a("The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues ")),s("mjx-container",k,[(l(),o("svg",m,e[6]||(e[6]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),e[7]||(e[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"λ")])],-1))]),e[13]||(e[13]=a(" describes a Lorentzian peak in the response; ")),s("mjx-container",g,[(l(),o("svg",u,e[8]||(e[8]=[i('',1)]))),e[9]||(e[9]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Re"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[14]||(e[14]=a(" gives its center and ")),s("mjx-container",b,[(l(),o("svg",y,e[10]||(e[10]=[i('',1)]))),e[11]||(e[11]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Im"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[15]||(e[15]=a(" its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment."))]),e[35]||(e[35]=s("p",null,"The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).",-1)),e[36]||(e[36]=s("p",null,[a("Behind the scenes, the spectra are stored using the dedicated structs "),s("code",null,"Lorentzian"),a(" and "),s("code",null,"JacobianSpectrum"),a(".")],-1)),s("details",f,[s("summary",null,[e[16]||(e[16]=s("a",{id:"HarmonicBalance.LinearResponse.JacobianSpectrum",href:"#HarmonicBalance.LinearResponse.JacobianSpectrum"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.JacobianSpectrum")],-1)),e[17]||(e[17]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[18]||(e[18]=i('
julia
mutable struct JacobianSpectrum
Holds a set of Lorentzian objects belonging to a variable.
Fields
  • peaks::Vector{HarmonicBalance.LinearResponse.Lorentzian}
Constructor
julia
JacobianSpectrum(res::Result; index::Int, branch::Int)
source
',7))]),s("details",T,[s("summary",null,[e[19]||(e[19]=s("a",{id:"HarmonicBalance.LinearResponse.Lorentzian",href:"#HarmonicBalance.LinearResponse.Lorentzian"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.Lorentzian")],-1)),e[20]||(e[20]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[21]||(e[21]=i('
julia
struct Lorentzian
Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).
Fields
  • ω0::Float64
  • Γ::Float64
  • A::Float64
source
',5))]),e[37]||(e[37]=s("h3",{id:"Higher-orders",tabindex:"-1"},[a("Higher orders "),s("a",{class:"header-anchor",href:"#Higher-orders","aria-label":'Permalink to "Higher orders {#Higher-orders}"'},"​")],-1)),e[38]||(e[38]=s("p",null,[a("Setting "),s("code",null,"order > 1"),a(" increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.")],-1)),s("details",Q,[s("summary",null,[e[22]||(e[22]=s("a",{id:"HarmonicBalance.LinearResponse.ResponseMatrix",href:"#HarmonicBalance.LinearResponse.ResponseMatrix"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.ResponseMatrix")],-1)),e[23]||(e[23]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[24]||(e[24]=i('
julia
struct ResponseMatrix
Holds the compiled response matrix of a system.
Fields
  • matrix::Matrix{Function}: The response matrix (compiled).
  • symbols::Vector{Num}: Any symbolic variables in matrix to be substituted at evaluation.
  • variables::Vector{HarmonicVariable}: The frequencies of the harmonic variables underlying matrix. These are needed to transform the harmonic variables to the non-rotating frame.
source
',5))]),s("details",x,[s("summary",null,[e[25]||(e[25]=s("a",{id:"HarmonicBalance.LinearResponse.get_response",href:"#HarmonicBalance.LinearResponse.get_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response")],-1)),e[26]||(e[26]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[27]||(e[27]=i(`
julia
get_response(
     rmat::HarmonicBalance.LinearResponse.ResponseMatrix,
     s::OrderedCollections.OrderedDict{Num, ComplexF64},
     Ω
-) -> Any
For rmat and a solution dictionary s, calculate the total response to a perturbative force at frequency Ω.
source
`,3))]),s("details",E,[s("summary",null,[e[28]||(e[28]=s("a",{id:"HarmonicBalance.LinearResponse.get_response_matrix",href:"#HarmonicBalance.LinearResponse.get_response_matrix"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response_matrix")],-1)),e[29]||(e[29]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[30]||(e[30]=i('
julia
get_response_matrix(diff_eq::DifferentialEquation, freq::Num; order=2)
Obtain the symbolic linear response matrix of a diff_eq corresponding to a perturbation frequency freq. This routine cannot accept a HarmonicEquation since there, some time-derivatives are already dropped. order denotes the highest differential order to be considered.
source
',3))])])}const V=r(d,[["render",H]]);export{B as __pageData,V as default};
+) -> Any

For rmat and a solution dictionary s, calculate the total response to a perturbative force at frequency Ω.

source

`,3))]),s("details",E,[s("summary",null,[e[28]||(e[28]=s("a",{id:"HarmonicBalance.LinearResponse.get_response_matrix",href:"#HarmonicBalance.LinearResponse.get_response_matrix"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response_matrix")],-1)),e[29]||(e[29]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[30]||(e[30]=i('
julia
get_response_matrix(diff_eq::DifferentialEquation, freq::Num; order=2)

Obtain the symbolic linear response matrix of a diff_eq corresponding to a perturbation frequency freq. This routine cannot accept a HarmonicEquation since there, some time-derivatives are already dropped. order denotes the highest differential order to be considered.

source

',3))])])}const V=r(d,[["render",H]]);export{B as __pageData,V as default}; diff --git a/dev/assets/manual_linear_response.md.CD1hVQFy.lean.js b/dev/assets/manual_linear_response.md.CafqEdgB.lean.js similarity index 95% rename from dev/assets/manual_linear_response.md.CD1hVQFy.lean.js rename to dev/assets/manual_linear_response.md.CafqEdgB.lean.js index 4b0231bb..b72c161c 100644 --- a/dev/assets/manual_linear_response.md.CD1hVQFy.lean.js +++ b/dev/assets/manual_linear_response.md.CafqEdgB.lean.js @@ -1,5 +1,5 @@ -import{_ as r,c as o,a4 as i,j as s,a,G as n,B as p,o as l}from"./chunks/framework.DGj8AcR1.js";const B=JSON.parse('{"title":"Linear response (WIP)","description":"","frontmatter":{},"headers":[],"relativePath":"manual/linear_response.md","filePath":"manual/linear_response.md"}'),d={name:"manual/linear_response.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},k={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.247ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2319 1000","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.278ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2333 1000","aria-hidden":"true"},f={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""};function H(v,e,j,w,L,F){const t=p("Badge");return l(),o("div",null,[e[31]||(e[31]=i('

Linear response (WIP) ​

This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.

The methodology used is explained in Jan Kosata phd thesis.

Stability ​

The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,

',5)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"HarmonicBalance.LinearResponse.get_Jacobian",href:"#HarmonicBalance.LinearResponse.get_Jacobian"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_Jacobian")],-1)),e[1]||(e[1]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=i('
julia
get_Jacobian(eom)

Obtain the symbolic Jacobian matrix of eom (either a HarmonicEquation or a DifferentialEquation). This is the linearised left-hand side of F(u) = du/dT.

source

Obtain a Jacobian from a DifferentialEquation by first converting it into a HarmonicEquation.

source

Get the Jacobian of a set of equations eqs with respect to the variables vars.

source

',7))]),e[32]||(e[32]=s("h2",{id:"Linear-response",tabindex:"-1"},[a("Linear response "),s("a",{class:"header-anchor",href:"#Linear-response","aria-label":'Permalink to "Linear response {#Linear-response}"'},"​")],-1)),e[33]||(e[33]=s("p",null,[a("The response to white noise can be shown with "),s("code",null,"plot_linear_response"),a(". Depending on the "),s("code",null,"order"),a(" argument, different methods are used.")],-1)),s("details",c,[s("summary",null,[e[3]||(e[3]=s("a",{id:"HarmonicBalance.LinearResponse.plot_linear_response",href:"#HarmonicBalance.LinearResponse.plot_linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_linear_response")],-1)),e[4]||(e[4]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[5]||(e[5]=i('
julia
plot_linear_response(res::Result, nat_var::Num; Ω_range, branch::Int, order=1, logscale=false, show_progress=true, kwargs...)

Plot the linear response to white noise of the variable nat_var for Result res on branch for input frequencies Ω_range. Slow-time derivatives up to order are kept in the process.

Any kwargs are fed to Plots' gr().

Solutions not belonging to the physical class are ignored.

source

',5))]),e[34]||(e[34]=s("h3",{id:"First-order",tabindex:"-1"},[a("First order "),s("a",{class:"header-anchor",href:"#First-order","aria-label":'Permalink to "First order {#First-order}"'},"​")],-1)),s("p",null,[e[12]||(e[12]=a("The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues ")),s("mjx-container",k,[(l(),o("svg",m,e[6]||(e[6]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),e[7]||(e[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"λ")])],-1))]),e[13]||(e[13]=a(" describes a Lorentzian peak in the response; ")),s("mjx-container",g,[(l(),o("svg",u,e[8]||(e[8]=[i('',1)]))),e[9]||(e[9]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Re"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[14]||(e[14]=a(" gives its center and ")),s("mjx-container",b,[(l(),o("svg",y,e[10]||(e[10]=[i('',1)]))),e[11]||(e[11]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Im"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[15]||(e[15]=a(" its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment."))]),e[35]||(e[35]=s("p",null,"The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).",-1)),e[36]||(e[36]=s("p",null,[a("Behind the scenes, the spectra are stored using the dedicated structs "),s("code",null,"Lorentzian"),a(" and "),s("code",null,"JacobianSpectrum"),a(".")],-1)),s("details",f,[s("summary",null,[e[16]||(e[16]=s("a",{id:"HarmonicBalance.LinearResponse.JacobianSpectrum",href:"#HarmonicBalance.LinearResponse.JacobianSpectrum"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.JacobianSpectrum")],-1)),e[17]||(e[17]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[18]||(e[18]=i('
julia
mutable struct JacobianSpectrum

Holds a set of Lorentzian objects belonging to a variable.

Fields

  • peaks::Vector{HarmonicBalance.LinearResponse.Lorentzian}

Constructor

julia
JacobianSpectrum(res::Result; index::Int, branch::Int)

source

',7))]),s("details",T,[s("summary",null,[e[19]||(e[19]=s("a",{id:"HarmonicBalance.LinearResponse.Lorentzian",href:"#HarmonicBalance.LinearResponse.Lorentzian"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.Lorentzian")],-1)),e[20]||(e[20]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[21]||(e[21]=i('
julia
struct Lorentzian

Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).

Fields

  • ω0::Float64

  • Γ::Float64

  • A::Float64

source

',5))]),e[37]||(e[37]=s("h3",{id:"Higher-orders",tabindex:"-1"},[a("Higher orders "),s("a",{class:"header-anchor",href:"#Higher-orders","aria-label":'Permalink to "Higher orders {#Higher-orders}"'},"​")],-1)),e[38]||(e[38]=s("p",null,[a("Setting "),s("code",null,"order > 1"),a(" increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.")],-1)),s("details",Q,[s("summary",null,[e[22]||(e[22]=s("a",{id:"HarmonicBalance.LinearResponse.ResponseMatrix",href:"#HarmonicBalance.LinearResponse.ResponseMatrix"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.ResponseMatrix")],-1)),e[23]||(e[23]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[24]||(e[24]=i('
julia
struct ResponseMatrix

Holds the compiled response matrix of a system.

Fields

  • matrix::Matrix{Function}: The response matrix (compiled).

  • symbols::Vector{Num}: Any symbolic variables in matrix to be substituted at evaluation.

  • variables::Vector{HarmonicVariable}: The frequencies of the harmonic variables underlying matrix. These are needed to transform the harmonic variables to the non-rotating frame.

source

',5))]),s("details",x,[s("summary",null,[e[25]||(e[25]=s("a",{id:"HarmonicBalance.LinearResponse.get_response",href:"#HarmonicBalance.LinearResponse.get_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response")],-1)),e[26]||(e[26]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[27]||(e[27]=i(`
julia
get_response(
+import{_ as r,c as o,a4 as i,j as s,a,G as n,B as p,o as l}from"./chunks/framework.DGj8AcR1.js";const B=JSON.parse('{"title":"Linear response (WIP)","description":"","frontmatter":{},"headers":[],"relativePath":"manual/linear_response.md","filePath":"manual/linear_response.md"}'),d={name:"manual/linear_response.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},k={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.247ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2319 1000","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.278ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2333 1000","aria-hidden":"true"},f={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""};function H(v,e,j,w,L,F){const t=p("Badge");return l(),o("div",null,[e[31]||(e[31]=i('
Linear response (WIP) ​
This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.
The methodology used is explained in Jan Kosata phd thesis.
Stability ​
The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,
',5)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"HarmonicBalance.LinearResponse.get_Jacobian",href:"#HarmonicBalance.LinearResponse.get_Jacobian"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_Jacobian")],-1)),e[1]||(e[1]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=i('
julia
get_Jacobian(eom)
Obtain the symbolic Jacobian matrix of eom (either a HarmonicEquation or a DifferentialEquation). This is the linearised left-hand side of F(u) = du/dT.
source
Obtain a Jacobian from a DifferentialEquation by first converting it into a HarmonicEquation.
source
Get the Jacobian of a set of equations eqs with respect to the variables vars.
source
',7))]),e[32]||(e[32]=s("h2",{id:"Linear-response",tabindex:"-1"},[a("Linear response "),s("a",{class:"header-anchor",href:"#Linear-response","aria-label":'Permalink to "Linear response {#Linear-response}"'},"​")],-1)),e[33]||(e[33]=s("p",null,[a("The response to white noise can be shown with "),s("code",null,"plot_linear_response"),a(". Depending on the "),s("code",null,"order"),a(" argument, different methods are used.")],-1)),s("details",c,[s("summary",null,[e[3]||(e[3]=s("a",{id:"HarmonicBalance.LinearResponse.plot_linear_response",href:"#HarmonicBalance.LinearResponse.plot_linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_linear_response")],-1)),e[4]||(e[4]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[5]||(e[5]=i('
julia
plot_linear_response(res::Result, nat_var::Num; Ω_range, branch::Int, order=1, logscale=false, show_progress=true, kwargs...)
Plot the linear response to white noise of the variable nat_var for Result res on branch for input frequencies Ω_range. Slow-time derivatives up to order are kept in the process.
Any kwargs are fed to Plots' gr().
Solutions not belonging to the physical class are ignored.
source
',5))]),e[34]||(e[34]=s("h3",{id:"First-order",tabindex:"-1"},[a("First order "),s("a",{class:"header-anchor",href:"#First-order","aria-label":'Permalink to "First order {#First-order}"'},"​")],-1)),s("p",null,[e[12]||(e[12]=a("The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues ")),s("mjx-container",k,[(l(),o("svg",m,e[6]||(e[6]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),e[7]||(e[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"λ")])],-1))]),e[13]||(e[13]=a(" describes a Lorentzian peak in the response; ")),s("mjx-container",g,[(l(),o("svg",u,e[8]||(e[8]=[i('',1)]))),e[9]||(e[9]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Re"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[14]||(e[14]=a(" gives its center and ")),s("mjx-container",b,[(l(),o("svg",y,e[10]||(e[10]=[i('',1)]))),e[11]||(e[11]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Im"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[15]||(e[15]=a(" its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment."))]),e[35]||(e[35]=s("p",null,"The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).",-1)),e[36]||(e[36]=s("p",null,[a("Behind the scenes, the spectra are stored using the dedicated structs "),s("code",null,"Lorentzian"),a(" and "),s("code",null,"JacobianSpectrum"),a(".")],-1)),s("details",f,[s("summary",null,[e[16]||(e[16]=s("a",{id:"HarmonicBalance.LinearResponse.JacobianSpectrum",href:"#HarmonicBalance.LinearResponse.JacobianSpectrum"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.JacobianSpectrum")],-1)),e[17]||(e[17]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[18]||(e[18]=i('
julia
mutable struct JacobianSpectrum
Holds a set of Lorentzian objects belonging to a variable.
Fields
  • peaks::Vector{HarmonicBalance.LinearResponse.Lorentzian}
Constructor
julia
JacobianSpectrum(res::Result; index::Int, branch::Int)
source
',7))]),s("details",T,[s("summary",null,[e[19]||(e[19]=s("a",{id:"HarmonicBalance.LinearResponse.Lorentzian",href:"#HarmonicBalance.LinearResponse.Lorentzian"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.Lorentzian")],-1)),e[20]||(e[20]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[21]||(e[21]=i('
julia
struct Lorentzian
Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).
Fields
  • ω0::Float64
  • Γ::Float64
  • A::Float64
source
',5))]),e[37]||(e[37]=s("h3",{id:"Higher-orders",tabindex:"-1"},[a("Higher orders "),s("a",{class:"header-anchor",href:"#Higher-orders","aria-label":'Permalink to "Higher orders {#Higher-orders}"'},"​")],-1)),e[38]||(e[38]=s("p",null,[a("Setting "),s("code",null,"order > 1"),a(" increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.")],-1)),s("details",Q,[s("summary",null,[e[22]||(e[22]=s("a",{id:"HarmonicBalance.LinearResponse.ResponseMatrix",href:"#HarmonicBalance.LinearResponse.ResponseMatrix"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.ResponseMatrix")],-1)),e[23]||(e[23]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[24]||(e[24]=i('
julia
struct ResponseMatrix
Holds the compiled response matrix of a system.
Fields
  • matrix::Matrix{Function}: The response matrix (compiled).
  • symbols::Vector{Num}: Any symbolic variables in matrix to be substituted at evaluation.
  • variables::Vector{HarmonicVariable}: The frequencies of the harmonic variables underlying matrix. These are needed to transform the harmonic variables to the non-rotating frame.
source
',5))]),s("details",x,[s("summary",null,[e[25]||(e[25]=s("a",{id:"HarmonicBalance.LinearResponse.get_response",href:"#HarmonicBalance.LinearResponse.get_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response")],-1)),e[26]||(e[26]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[27]||(e[27]=i(`
julia
get_response(
     rmat::HarmonicBalance.LinearResponse.ResponseMatrix,
     s::OrderedCollections.OrderedDict{Num, ComplexF64},
     Ω
-) -> Any
For rmat and a solution dictionary s, calculate the total response to a perturbative force at frequency Ω.
source
`,3))]),s("details",E,[s("summary",null,[e[28]||(e[28]=s("a",{id:"HarmonicBalance.LinearResponse.get_response_matrix",href:"#HarmonicBalance.LinearResponse.get_response_matrix"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response_matrix")],-1)),e[29]||(e[29]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[30]||(e[30]=i('
julia
get_response_matrix(diff_eq::DifferentialEquation, freq::Num; order=2)
Obtain the symbolic linear response matrix of a diff_eq corresponding to a perturbation frequency freq. This routine cannot accept a HarmonicEquation since there, some time-derivatives are already dropped. order denotes the highest differential order to be considered.
source
',3))])])}const V=r(d,[["render",H]]);export{B as __pageData,V as default};
+) -> Any

For rmat and a solution dictionary s, calculate the total response to a perturbative force at frequency Ω.

source

`,3))]),s("details",E,[s("summary",null,[e[28]||(e[28]=s("a",{id:"HarmonicBalance.LinearResponse.get_response_matrix",href:"#HarmonicBalance.LinearResponse.get_response_matrix"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response_matrix")],-1)),e[29]||(e[29]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[30]||(e[30]=i('
julia
get_response_matrix(diff_eq::DifferentialEquation, freq::Num; order=2)

Obtain the symbolic linear response matrix of a diff_eq corresponding to a perturbation frequency freq. This routine cannot accept a HarmonicEquation since there, some time-derivatives are already dropped. order denotes the highest differential order to be considered.

source

',3))])])}const V=r(d,[["render",H]]);export{B as __pageData,V as default}; diff --git a/dev/assets/manual_methods.md.DTOoMn0y.js b/dev/assets/manual_methods.md.DTOoMn0y.js new file mode 100644 index 00000000..447ad7c6 --- /dev/null +++ b/dev/assets/manual_methods.md.DTOoMn0y.js @@ -0,0 +1 @@ +import{_ as i,c as l,j as t,a,G as s,a4 as o,B as r,o as n}from"./chunks/framework.DGj8AcR1.js";const v=JSON.parse('{"title":"Methods","description":"","frontmatter":{},"headers":[],"relativePath":"manual/methods.md","filePath":"manual/methods.md"}'),T={name:"manual/methods.md"},d={class:"jldocstring custom-block",open:""},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"30.769ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 13600.1 1000","aria-hidden":"true"},h={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"17.717ex",height:"2.587ex",role:"img",focusable:"false",viewBox:"0 -893.3 7831 1143.3","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.452ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.534ex",height:"2.149ex",role:"img",focusable:"false",viewBox:"0 -750 1120 950","aria-hidden":"true"},f={class:"jldocstring custom-block",open:""},H={class:"jldocstring custom-block",open:""};function y(x,e,w,b,k,M){const Q=r("Badge");return n(),l("div",null,[e[24]||(e[24]=t("h1",{id:"methods",tabindex:"-1"},[a("Methods "),t("a",{class:"header-anchor",href:"#methods","aria-label":'Permalink to "Methods"'},"​")],-1)),e[25]||(e[25]=t("p",null,"We offer several methods for solving the nonlinear algebraic equations that arise from the harmonic balance procedure. Each method has different tradeoffs between speed, robustness, and completeness.",-1)),e[26]||(e[26]=t("h2",{id:"Total-Degree-Method",tabindex:"-1"},[a("Total Degree Method "),t("a",{class:"header-anchor",href:"#Total-Degree-Method","aria-label":'Permalink to "Total Degree Method {#Total-Degree-Method}"'},"​")],-1)),t("details",d,[t("summary",null,[e[0]||(e[0]=t("a",{id:"HarmonicBalance.TotalDegree",href:"#HarmonicBalance.TotalDegree"},[t("span",{class:"jlbinding"},"HarmonicBalance.TotalDegree")],-1)),e[1]||(e[1]=a()),s(Q,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[16]||(e[16]=o('
julia
TotalDegree
',1)),t("p",null,[e[8]||(e[8]=a("The Total Degree homotopy method performs a homotopy ")),t("mjx-container",m,[(n(),l("svg",p,e[2]||(e[2]=[o('',1)]))),e[3]||(e[3]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"H"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",null,","),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"γ"),t("mi",null,"t"),t("mi",null,"G"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",null,"−"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mi",null,"F"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")")])],-1))]),e[9]||(e[9]=a(" from the trivial polynomial system ")),t("mjx-container",h,[(n(),l("svg",c,e[4]||(e[4]=[o('ᵢᵢᵢ',1)]))),e[5]||(e[5]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"F"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"x"),t("msup",null,[t("mi",null,"ᵢ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"d"),t("mi",null,"ᵢ")])]),t("mo",null,"+"),t("mi",null,"a"),t("mi",null,"ᵢ")])],-1))]),e[10]||(e[10]=a(" with the maximal degree ")),t("mjx-container",g,[(n(),l("svg",u,e[6]||(e[6]=[o('ᵢ',1)]))),e[7]||(e[7]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"d"),t("mi",null,"ᵢ")])],-1))]),e[11]||(e[11]=a(" determined by the ")),e[12]||(e[12]=t("a",{href:"https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem",target:"_blank",rel:"noreferrer"},"Bezout bound",-1)),e[13]||(e[13]=a(". The method guarantees to find all solutions, however, it comes with a high computational cost. See ")),e[14]||(e[14]=t("a",{href:"https://www.juliahomotopycontinuation.org/guides/totaldegree/",target:"_blank",rel:"noreferrer"},"HomotopyContinuation.jl",-1)),e[15]||(e[15]=a(" for more information."))]),e[17]||(e[17]=o('

Fields

  • gamma::Complex: Complex multiplying factor of the start system G(x) for the homotopy

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',3))]),e[27]||(e[27]=t("h2",{id:"Polyhedral-Method",tabindex:"-1"},[a("Polyhedral Method "),t("a",{class:"header-anchor",href:"#Polyhedral-Method","aria-label":'Permalink to "Polyhedral Method {#Polyhedral-Method}"'},"​")],-1)),t("details",f,[t("summary",null,[e[18]||(e[18]=t("a",{id:"HarmonicBalance.Polyhedral",href:"#HarmonicBalance.Polyhedral"},[t("span",{class:"jlbinding"},"HarmonicBalance.Polyhedral")],-1)),e[19]||(e[19]=a()),s(Q,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[20]||(e[20]=o('
julia
Polyhedral

The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (only_non_zero = true), resulting in a speed up. See HomotopyContinuation.jl for more information.

Fields

  • only_non_zero::Bool: Boolean indicating if only non-zero solutions are considered.

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',5))]),e[28]||(e[28]=t("h2",{id:"Warm-Up-Method",tabindex:"-1"},[a("Warm Up Method "),t("a",{class:"header-anchor",href:"#Warm-Up-Method","aria-label":'Permalink to "Warm Up Method {#Warm-Up-Method}"'},"​")],-1)),t("details",H,[t("summary",null,[e[21]||(e[21]=t("a",{id:"HarmonicBalance.WarmUp",href:"#HarmonicBalance.WarmUp"},[t("span",{class:"jlbinding"},"HarmonicBalance.WarmUp")],-1)),e[22]||(e[22]=a()),s(Q,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[23]||(e[23]=o('
julia
WarmUp

The Warm Up method prepares a warmup system with the Total Degree method using the parameter at index perturbed by perturbation_size. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method does not guarantee to find all solutions. See HomotopyContinuation.jl for more information.

Fields

  • perturbation_size::ComplexF64: Size of the perturbation.

  • index::Union{Int64, EndpointRanges.Endpoint}: Index for the parameter set used as start system.

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',5))])])}const j=i(T,[["render",y]]);export{v as __pageData,j as default}; diff --git a/dev/assets/manual_methods.md.DTOoMn0y.lean.js b/dev/assets/manual_methods.md.DTOoMn0y.lean.js new file mode 100644 index 00000000..447ad7c6 --- /dev/null +++ b/dev/assets/manual_methods.md.DTOoMn0y.lean.js @@ -0,0 +1 @@ +import{_ as i,c as l,j as t,a,G as s,a4 as o,B as r,o as n}from"./chunks/framework.DGj8AcR1.js";const v=JSON.parse('{"title":"Methods","description":"","frontmatter":{},"headers":[],"relativePath":"manual/methods.md","filePath":"manual/methods.md"}'),T={name:"manual/methods.md"},d={class:"jldocstring custom-block",open:""},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"30.769ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 13600.1 1000","aria-hidden":"true"},h={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"17.717ex",height:"2.587ex",role:"img",focusable:"false",viewBox:"0 -893.3 7831 1143.3","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.452ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.534ex",height:"2.149ex",role:"img",focusable:"false",viewBox:"0 -750 1120 950","aria-hidden":"true"},f={class:"jldocstring custom-block",open:""},H={class:"jldocstring custom-block",open:""};function y(x,e,w,b,k,M){const Q=r("Badge");return n(),l("div",null,[e[24]||(e[24]=t("h1",{id:"methods",tabindex:"-1"},[a("Methods "),t("a",{class:"header-anchor",href:"#methods","aria-label":'Permalink to "Methods"'},"​")],-1)),e[25]||(e[25]=t("p",null,"We offer several methods for solving the nonlinear algebraic equations that arise from the harmonic balance procedure. Each method has different tradeoffs between speed, robustness, and completeness.",-1)),e[26]||(e[26]=t("h2",{id:"Total-Degree-Method",tabindex:"-1"},[a("Total Degree Method "),t("a",{class:"header-anchor",href:"#Total-Degree-Method","aria-label":'Permalink to "Total Degree Method {#Total-Degree-Method}"'},"​")],-1)),t("details",d,[t("summary",null,[e[0]||(e[0]=t("a",{id:"HarmonicBalance.TotalDegree",href:"#HarmonicBalance.TotalDegree"},[t("span",{class:"jlbinding"},"HarmonicBalance.TotalDegree")],-1)),e[1]||(e[1]=a()),s(Q,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[16]||(e[16]=o('
julia
TotalDegree
',1)),t("p",null,[e[8]||(e[8]=a("The Total Degree homotopy method performs a homotopy ")),t("mjx-container",m,[(n(),l("svg",p,e[2]||(e[2]=[o('',1)]))),e[3]||(e[3]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"H"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",null,","),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"γ"),t("mi",null,"t"),t("mi",null,"G"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",null,"−"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mi",null,"F"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")")])],-1))]),e[9]||(e[9]=a(" from the trivial polynomial system ")),t("mjx-container",h,[(n(),l("svg",c,e[4]||(e[4]=[o('ᵢᵢᵢ',1)]))),e[5]||(e[5]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"F"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"x"),t("msup",null,[t("mi",null,"ᵢ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"d"),t("mi",null,"ᵢ")])]),t("mo",null,"+"),t("mi",null,"a"),t("mi",null,"ᵢ")])],-1))]),e[10]||(e[10]=a(" with the maximal degree ")),t("mjx-container",g,[(n(),l("svg",u,e[6]||(e[6]=[o('ᵢ',1)]))),e[7]||(e[7]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"d"),t("mi",null,"ᵢ")])],-1))]),e[11]||(e[11]=a(" determined by the ")),e[12]||(e[12]=t("a",{href:"https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem",target:"_blank",rel:"noreferrer"},"Bezout bound",-1)),e[13]||(e[13]=a(". The method guarantees to find all solutions, however, it comes with a high computational cost. See ")),e[14]||(e[14]=t("a",{href:"https://www.juliahomotopycontinuation.org/guides/totaldegree/",target:"_blank",rel:"noreferrer"},"HomotopyContinuation.jl",-1)),e[15]||(e[15]=a(" for more information."))]),e[17]||(e[17]=o('

Fields

  • gamma::Complex: Complex multiplying factor of the start system G(x) for the homotopy

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',3))]),e[27]||(e[27]=t("h2",{id:"Polyhedral-Method",tabindex:"-1"},[a("Polyhedral Method "),t("a",{class:"header-anchor",href:"#Polyhedral-Method","aria-label":'Permalink to "Polyhedral Method {#Polyhedral-Method}"'},"​")],-1)),t("details",f,[t("summary",null,[e[18]||(e[18]=t("a",{id:"HarmonicBalance.Polyhedral",href:"#HarmonicBalance.Polyhedral"},[t("span",{class:"jlbinding"},"HarmonicBalance.Polyhedral")],-1)),e[19]||(e[19]=a()),s(Q,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[20]||(e[20]=o('
julia
Polyhedral

The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (only_non_zero = true), resulting in a speed up. See HomotopyContinuation.jl for more information.

Fields

  • only_non_zero::Bool: Boolean indicating if only non-zero solutions are considered.

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',5))]),e[28]||(e[28]=t("h2",{id:"Warm-Up-Method",tabindex:"-1"},[a("Warm Up Method "),t("a",{class:"header-anchor",href:"#Warm-Up-Method","aria-label":'Permalink to "Warm Up Method {#Warm-Up-Method}"'},"​")],-1)),t("details",H,[t("summary",null,[e[21]||(e[21]=t("a",{id:"HarmonicBalance.WarmUp",href:"#HarmonicBalance.WarmUp"},[t("span",{class:"jlbinding"},"HarmonicBalance.WarmUp")],-1)),e[22]||(e[22]=a()),s(Q,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[23]||(e[23]=o('
julia
WarmUp

The Warm Up method prepares a warmup system with the Total Degree method using the parameter at index perturbed by perturbation_size. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method does not guarantee to find all solutions. See HomotopyContinuation.jl for more information.

Fields

  • perturbation_size::ComplexF64: Size of the perturbation.

  • index::Union{Int64, EndpointRanges.Endpoint}: Index for the parameter set used as start system.

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',5))])])}const j=i(T,[["render",y]]);export{v as __pageData,j as default}; diff --git a/dev/assets/manual_methods.md.qGkkhEfO.js b/dev/assets/manual_methods.md.qGkkhEfO.js deleted file mode 100644 index fe190d3d..00000000 --- a/dev/assets/manual_methods.md.qGkkhEfO.js +++ /dev/null @@ -1 +0,0 @@ -import{_ as r,c as l,j as t,a as o,G as i,a4 as a,B as Q,o as n}from"./chunks/framework.DGj8AcR1.js";const v=JSON.parse('{"title":"Methods","description":"","frontmatter":{},"headers":[],"relativePath":"manual/methods.md","filePath":"manual/methods.md"}'),d={name:"manual/methods.md"},T={class:"jldocstring custom-block",open:""},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"30.769ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 13600.1 1000","aria-hidden":"true"},h={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.452ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.951ex",height:"2.474ex",role:"img",focusable:"false",viewBox:"0 -893.3 4398.4 1093.3","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},f={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.452ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.534ex",height:"2.149ex",role:"img",focusable:"false",viewBox:"0 -750 1120 950","aria-hidden":"true"},u={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""};function H(x,e,w,b,k,M){const s=Q("Badge");return n(),l("div",null,[e[24]||(e[24]=t("h1",{id:"methods",tabindex:"-1"},[o("Methods "),t("a",{class:"header-anchor",href:"#methods","aria-label":'Permalink to "Methods"'},"​")],-1)),e[25]||(e[25]=t("p",null,"We offer several methods for solving the nonlinear algebraic equations that arise from the harmonic balance procedure. Each method has different tradeoffs between speed, robustness, and completeness.",-1)),e[26]||(e[26]=t("h2",{id:"Total-Degree-Method",tabindex:"-1"},[o("Total Degree Method "),t("a",{class:"header-anchor",href:"#Total-Degree-Method","aria-label":'Permalink to "Total Degree Method {#Total-Degree-Method}"'},"​")],-1)),t("details",T,[t("summary",null,[e[0]||(e[0]=t("a",{id:"HarmonicBalance.TotalDegree",href:"#HarmonicBalance.TotalDegree"},[t("span",{class:"jlbinding"},"HarmonicBalance.TotalDegree")],-1)),e[1]||(e[1]=o()),i(s,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[16]||(e[16]=a('
julia
TotalDegree
',1)),t("p",null,[e[8]||(e[8]=o("The Total Degree homotopy method. Performs a homotopy ")),t("mjx-container",m,[(n(),l("svg",p,e[2]||(e[2]=[a('',1)]))),e[3]||(e[3]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"H"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",null,","),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"γ"),t("mi",null,"t"),t("mi",null,"G"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",null,"−"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mi",null,"F"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")")])],-1))]),e[9]||(e[9]=o(" from the trivial polynomial system ")),t("mjx-container",h,[(n(),l("svg",c,e[4]||(e[4]=[a('ᵢᵢᵢ',1)]))),e[5]||(e[5]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("msup",null,[t("mi",null,"ᵢ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"d"),t("mi",null,"ᵢ")])]),t("mo",null,"+"),t("mi",null,"a"),t("mi",null,"ᵢ")])],-1))]),e[10]||(e[10]=o(" with the maximal degree ")),t("mjx-container",g,[(n(),l("svg",f,e[6]||(e[6]=[a('ᵢ',1)]))),e[7]||(e[7]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"d"),t("mi",null,"ᵢ")])],-1))]),e[11]||(e[11]=o(" determined by the ")),e[12]||(e[12]=t("a",{href:"https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem",target:"_blank",rel:"noreferrer"},"Bezout bound",-1)),e[13]||(e[13]=o(". The method guarantees to find all solutions, however, it comes with a high computational cost. See ")),e[14]||(e[14]=t("a",{href:"https://www.juliahomotopycontinuation.org/guides/totaldegree/",target:"_blank",rel:"noreferrer"},"HomotopyContinuation.jl",-1)),e[15]||(e[15]=o(" for more information."))]),e[17]||(e[17]=a('

Fields

  • gamma::Complex: Complex multiplying factor of the start system G(x) for the homotopy

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',3))]),e[27]||(e[27]=t("h2",{id:"Polyhedral-Method",tabindex:"-1"},[o("Polyhedral Method "),t("a",{class:"header-anchor",href:"#Polyhedral-Method","aria-label":'Permalink to "Polyhedral Method {#Polyhedral-Method}"'},"​")],-1)),t("details",u,[t("summary",null,[e[18]||(e[18]=t("a",{id:"HarmonicBalance.Polyhedral",href:"#HarmonicBalance.Polyhedral"},[t("span",{class:"jlbinding"},"HarmonicBalance.Polyhedral")],-1)),e[19]||(e[19]=o()),i(s,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[20]||(e[20]=a('
julia
Polyhedral

The Polyhedral homotopy method. This method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (only_non_zero = true), resulting in speed up. See HomotopyContinuation.jl for more information.

Fields

  • only_non_zero::Bool: Boolean indicating if only non-zero solutions are considered.

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',5))]),e[28]||(e[28]=t("h2",{id:"Warm-Up-Method",tabindex:"-1"},[o("Warm Up Method "),t("a",{class:"header-anchor",href:"#Warm-Up-Method","aria-label":'Permalink to "Warm Up Method {#Warm-Up-Method}"'},"​")],-1)),t("details",y,[t("summary",null,[e[21]||(e[21]=t("a",{id:"HarmonicBalance.WarmUp",href:"#HarmonicBalance.WarmUp"},[t("span",{class:"jlbinding"},"HarmonicBalance.WarmUp")],-1)),e[22]||(e[22]=o()),i(s,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[23]||(e[23]=a('
julia
WarmUp

The Warm Up method. This method prepares a warmup system using the parameter at index perturbed by perturbation_size and performs a homotopy using the warmup system to all other systems in the parameter sweep. It is very efficient for systems with less bifurcation in the parameter sweep. The Warm Up method does not guarantee to find all solutions. See HomotopyContinuation.jl for more information.

Fields

  • perturbation_size::ComplexF64: Size of the perturbation.

  • index::Union{Int64, EndpointRanges.Endpoint}: Index for the endpoint.

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',5))])])}const j=r(d,[["render",H]]);export{v as __pageData,j as default}; diff --git a/dev/assets/manual_methods.md.qGkkhEfO.lean.js b/dev/assets/manual_methods.md.qGkkhEfO.lean.js deleted file mode 100644 index fe190d3d..00000000 --- a/dev/assets/manual_methods.md.qGkkhEfO.lean.js +++ /dev/null @@ -1 +0,0 @@ -import{_ as r,c as l,j as t,a as o,G as i,a4 as a,B as Q,o as n}from"./chunks/framework.DGj8AcR1.js";const v=JSON.parse('{"title":"Methods","description":"","frontmatter":{},"headers":[],"relativePath":"manual/methods.md","filePath":"manual/methods.md"}'),d={name:"manual/methods.md"},T={class:"jldocstring custom-block",open:""},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"30.769ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 13600.1 1000","aria-hidden":"true"},h={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.452ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.951ex",height:"2.474ex",role:"img",focusable:"false",viewBox:"0 -893.3 4398.4 1093.3","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},f={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.452ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.534ex",height:"2.149ex",role:"img",focusable:"false",viewBox:"0 -750 1120 950","aria-hidden":"true"},u={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""};function H(x,e,w,b,k,M){const s=Q("Badge");return n(),l("div",null,[e[24]||(e[24]=t("h1",{id:"methods",tabindex:"-1"},[o("Methods "),t("a",{class:"header-anchor",href:"#methods","aria-label":'Permalink to "Methods"'},"​")],-1)),e[25]||(e[25]=t("p",null,"We offer several methods for solving the nonlinear algebraic equations that arise from the harmonic balance procedure. Each method has different tradeoffs between speed, robustness, and completeness.",-1)),e[26]||(e[26]=t("h2",{id:"Total-Degree-Method",tabindex:"-1"},[o("Total Degree Method "),t("a",{class:"header-anchor",href:"#Total-Degree-Method","aria-label":'Permalink to "Total Degree Method {#Total-Degree-Method}"'},"​")],-1)),t("details",T,[t("summary",null,[e[0]||(e[0]=t("a",{id:"HarmonicBalance.TotalDegree",href:"#HarmonicBalance.TotalDegree"},[t("span",{class:"jlbinding"},"HarmonicBalance.TotalDegree")],-1)),e[1]||(e[1]=o()),i(s,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[16]||(e[16]=a('
julia
TotalDegree
',1)),t("p",null,[e[8]||(e[8]=o("The Total Degree homotopy method. Performs a homotopy ")),t("mjx-container",m,[(n(),l("svg",p,e[2]||(e[2]=[a('',1)]))),e[3]||(e[3]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"H"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",null,","),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"γ"),t("mi",null,"t"),t("mi",null,"G"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",null,"−"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mi",null,"F"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")")])],-1))]),e[9]||(e[9]=o(" from the trivial polynomial system ")),t("mjx-container",h,[(n(),l("svg",c,e[4]||(e[4]=[a('ᵢᵢᵢ',1)]))),e[5]||(e[5]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("msup",null,[t("mi",null,"ᵢ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"d"),t("mi",null,"ᵢ")])]),t("mo",null,"+"),t("mi",null,"a"),t("mi",null,"ᵢ")])],-1))]),e[10]||(e[10]=o(" with the maximal degree ")),t("mjx-container",g,[(n(),l("svg",f,e[6]||(e[6]=[a('ᵢ',1)]))),e[7]||(e[7]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"d"),t("mi",null,"ᵢ")])],-1))]),e[11]||(e[11]=o(" determined by the ")),e[12]||(e[12]=t("a",{href:"https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem",target:"_blank",rel:"noreferrer"},"Bezout bound",-1)),e[13]||(e[13]=o(". The method guarantees to find all solutions, however, it comes with a high computational cost. See ")),e[14]||(e[14]=t("a",{href:"https://www.juliahomotopycontinuation.org/guides/totaldegree/",target:"_blank",rel:"noreferrer"},"HomotopyContinuation.jl",-1)),e[15]||(e[15]=o(" for more information."))]),e[17]||(e[17]=a('

Fields

  • gamma::Complex: Complex multiplying factor of the start system G(x) for the homotopy

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',3))]),e[27]||(e[27]=t("h2",{id:"Polyhedral-Method",tabindex:"-1"},[o("Polyhedral Method "),t("a",{class:"header-anchor",href:"#Polyhedral-Method","aria-label":'Permalink to "Polyhedral Method {#Polyhedral-Method}"'},"​")],-1)),t("details",u,[t("summary",null,[e[18]||(e[18]=t("a",{id:"HarmonicBalance.Polyhedral",href:"#HarmonicBalance.Polyhedral"},[t("span",{class:"jlbinding"},"HarmonicBalance.Polyhedral")],-1)),e[19]||(e[19]=o()),i(s,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[20]||(e[20]=a('
julia
Polyhedral

The Polyhedral homotopy method. This method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (only_non_zero = true), resulting in speed up. See HomotopyContinuation.jl for more information.

Fields

  • only_non_zero::Bool: Boolean indicating if only non-zero solutions are considered.

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',5))]),e[28]||(e[28]=t("h2",{id:"Warm-Up-Method",tabindex:"-1"},[o("Warm Up Method "),t("a",{class:"header-anchor",href:"#Warm-Up-Method","aria-label":'Permalink to "Warm Up Method {#Warm-Up-Method}"'},"​")],-1)),t("details",y,[t("summary",null,[e[21]||(e[21]=t("a",{id:"HarmonicBalance.WarmUp",href:"#HarmonicBalance.WarmUp"},[t("span",{class:"jlbinding"},"HarmonicBalance.WarmUp")],-1)),e[22]||(e[22]=o()),i(s,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[23]||(e[23]=a('
julia
WarmUp

The Warm Up method. This method prepares a warmup system using the parameter at index perturbed by perturbation_size and performs a homotopy using the warmup system to all other systems in the parameter sweep. It is very efficient for systems with less bifurcation in the parameter sweep. The Warm Up method does not guarantee to find all solutions. See HomotopyContinuation.jl for more information.

Fields

  • perturbation_size::ComplexF64: Size of the perturbation.

  • index::Union{Int64, EndpointRanges.Endpoint}: Index for the endpoint.

  • thread::Bool: Boolean indicating if threading is enabled.

  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.

  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.

  • compile::Union{Bool, Symbol}: Compilation options.

  • seed::UInt32: Seed for random number generation.

source

',5))])])}const j=r(d,[["render",H]]);export{v as __pageData,j as default}; diff --git a/dev/assets/manual_plotting.md.CMWmhlSv.js b/dev/assets/manual_plotting.md.C0lzz6zg.js similarity index 92% rename from dev/assets/manual_plotting.md.CMWmhlSv.js rename to dev/assets/manual_plotting.md.C0lzz6zg.js index d52c02c6..e837e898 100644 --- a/dev/assets/manual_plotting.md.CMWmhlSv.js +++ b/dev/assets/manual_plotting.md.C0lzz6zg.js @@ -1,18 +1,18 @@ -import{_ as l,c as o,j as a,a as t,G as e,a4 as n,B as p,o as r}from"./chunks/framework.DGj8AcR1.js";const j=JSON.parse('{"title":"Analysis and plotting","description":"","frontmatter":{},"headers":[],"relativePath":"manual/plotting.md","filePath":"manual/plotting.md"}'),d={name:"manual/plotting.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""};function u(y,s,m,b,E,f){const i=p("Badge");return r(),o("div",null,[s[12]||(s[12]=a("h1",{id:"Analysis-and-plotting",tabindex:"-1"},[t("Analysis and plotting "),a("a",{class:"header-anchor",href:"#Analysis-and-plotting","aria-label":'Permalink to "Analysis and plotting {#Analysis-and-plotting}"'},"​")],-1)),s[13]||(s[13]=a("p",null,[t("The key method for visualization is "),a("code",null,"transform_solutions"),t(", which parses a string into a symbolic expression and evaluates it for every steady state solution.")],-1)),a("details",h,[a("summary",null,[s[0]||(s[0]=a("a",{id:"HarmonicBalance.transform_solutions",href:"#HarmonicBalance.transform_solutions"},[a("span",{class:"jlbinding"},"HarmonicBalance.transform_solutions")],-1)),s[1]||(s[1]=t()),e(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=n(`
julia
transform_solutions(
+import{_ as l,c as o,j as a,a as t,G as n,a4 as e,B as p,o as r}from"./chunks/framework.DGj8AcR1.js";const j=JSON.parse('{"title":"Analysis and plotting","description":"","frontmatter":{},"headers":[],"relativePath":"manual/plotting.md","filePath":"manual/plotting.md"}'),d={name:"manual/plotting.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""};function u(y,s,m,b,E,f){const i=p("Badge");return r(),o("div",null,[s[12]||(s[12]=a("h1",{id:"Analysis-and-plotting",tabindex:"-1"},[t("Analysis and plotting "),a("a",{class:"header-anchor",href:"#Analysis-and-plotting","aria-label":'Permalink to "Analysis and plotting {#Analysis-and-plotting}"'},"​")],-1)),s[13]||(s[13]=a("p",null,[t("The key method for visualization is "),a("code",null,"transform_solutions"),t(", which parses a string into a symbolic expression and evaluates it for every steady state solution.")],-1)),a("details",h,[a("summary",null,[s[0]||(s[0]=a("a",{id:"HarmonicBalance.transform_solutions",href:"#HarmonicBalance.transform_solutions"},[a("span",{class:"jlbinding"},"HarmonicBalance.transform_solutions")],-1)),s[1]||(s[1]=t()),n(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=e(`
julia
transform_solutions(
     res::HarmonicBalance.Result,
     func;
     branches,
     realify
-) -> Vector
Takes a Result object and a string f representing a Symbolics.jl expression. Returns an array with the values of f evaluated for the respective solutions. Additional substitution rules can be specified in rules in the format ("a" => val) or (a => val)
source
`,3))]),s[14]||(s[14]=a("h2",{id:"Plotting-solutions",tabindex:"-1"},[t("Plotting solutions "),a("a",{class:"header-anchor",href:"#Plotting-solutions","aria-label":'Permalink to "Plotting solutions {#Plotting-solutions}"'},"​")],-1)),s[15]||(s[15]=a("p",null,[t("The function "),a("code",null,"plot"),t(" is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by "),a("code",null,"sort_solutions"),t(".")],-1)),a("details",c,[a("summary",null,[s[3]||(s[3]=a("a",{id:"RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}",href:"#RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}"},[a("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[4]||(s[4]=t()),e(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[5]||(s[5]=n(`
julia
plot(
+) -> Vector
Takes a Result object and a string f representing a Symbolics.jl expression. Returns an array with the values of f evaluated for the respective solutions. Additional substitution rules can be specified in rules in the format ("a" => val) or (a => val)
source
`,3))]),s[14]||(s[14]=a("h2",{id:"Plotting-solutions",tabindex:"-1"},[t("Plotting solutions "),a("a",{class:"header-anchor",href:"#Plotting-solutions","aria-label":'Permalink to "Plotting solutions {#Plotting-solutions}"'},"​")],-1)),s[15]||(s[15]=a("p",null,[t("The function "),a("code",null,"plot"),t(" is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by "),a("code",null,"sort_solutions"),t(".")],-1)),a("details",c,[a("summary",null,[s[3]||(s[3]=a("a",{id:"RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}",href:"#RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}"},[a("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[4]||(s[4]=t()),n(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[5]||(s[5]=e(`
julia
plot(
     res::HarmonicBalance.Result,
     varargs...;
     cut,
     kwargs...
 ) -> Plots.Plot
Plot a Result object.
Class selection done by passing String or Vector{String} as kwarg:
class       :   only plot solutions in this class(es) ("all" --> plot everything)
 not_class   :   do not plot solutions in this class(es)
Other kwargs are passed onto Plots.gr().
See also plot!
The x,y,z arguments are Strings compatible with Symbolics.jl, e.g., y=2*sqrt(u1^2+v1^2) plots the amplitude of the first quadratures multiplied by 2.
1D plots
plot(res::Result; x::String, y::String, class="default", not_class=[], kwargs...)
-plot(res::Result, y::String; kwargs...) # take x automatically from Result
Default behaviour is to plot stable solutions as full lines, unstable as dashed.
If a sweep in two parameters were done, i.e., dim(res)==2, a one dimensional cut can be plotted by using the keyword cut were it takes a Pair{Num, Float64} type entry. For example, plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2)) plots a cut at λ = 0.2.
2D plots
plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)
To make the 2d plot less chaotic it is required to specify the specific branch to plot, labeled by a Int64.
The x and y axes are taken automatically from res
source
`,17))]),s[16]||(s[16]=a("h2",{id:"Plotting-phase-diagrams",tabindex:"-1"},[t("Plotting phase diagrams "),a("a",{class:"header-anchor",href:"#Plotting-phase-diagrams","aria-label":'Permalink to "Plotting phase diagrams {#Plotting-phase-diagrams}"'},"​")],-1)),s[17]||(s[17]=a("p",null,[t("In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. "),a("code",null,"plot_phase_diagram"),t(" handles this for 1D and 2D datasets.")],-1)),a("details",g,[a("summary",null,[s[6]||(s[6]=a("a",{id:"HarmonicBalance.plot_phase_diagram",href:"#HarmonicBalance.plot_phase_diagram"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_phase_diagram")],-1)),s[7]||(s[7]=t()),e(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[8]||(s[8]=n(`
julia
plot_phase_diagram(
+plot(res::Result, y::String; kwargs...) # take x automatically from Result
Default behaviour is to plot stable solutions as full lines, unstable as dashed.
If a sweep in two parameters were done, i.e., dim(res)==2, a one dimensional cut can be plotted by using the keyword cut were it takes a Pair{Num, Float64} type entry. For example, plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2)) plots a cut at λ = 0.2.
2D plots
plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)
To make the 2d plot less chaotic it is required to specify the specific branch to plot, labeled by a Int64.
The x and y axes are taken automatically from res
source
`,17))]),s[16]||(s[16]=a("h2",{id:"Plotting-phase-diagrams",tabindex:"-1"},[t("Plotting phase diagrams "),a("a",{class:"header-anchor",href:"#Plotting-phase-diagrams","aria-label":'Permalink to "Plotting phase diagrams {#Plotting-phase-diagrams}"'},"​")],-1)),s[17]||(s[17]=a("p",null,[t("In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. "),a("code",null,"plot_phase_diagram"),t(" handles this for 1D and 2D datasets.")],-1)),a("details",g,[a("summary",null,[s[6]||(s[6]=a("a",{id:"HarmonicBalance.plot_phase_diagram",href:"#HarmonicBalance.plot_phase_diagram"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_phase_diagram")],-1)),s[7]||(s[7]=t()),n(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[8]||(s[8]=e(`
julia
plot_phase_diagram(
     res::HarmonicBalance.Result;
     kwargs...
 ) -> Plots.Plot
Plot the number of solutions in a Result object as a function of the parameters. Works with 1D and 2D datasets.
Class selection done by passing String or Vector{String} as kwarg:
class::String       :   only count solutions in this class ("all" --> plot everything)
-not_class::String   :   do not count solutions in this class
Other kwargs are passed onto Plots.gr()
source
`,6))]),s[18]||(s[18]=a("h2",{id:"Plot-spaghetti-plot",tabindex:"-1"},[t("Plot spaghetti plot "),a("a",{class:"header-anchor",href:"#Plot-spaghetti-plot","aria-label":'Permalink to "Plot spaghetti plot {#Plot-spaghetti-plot}"'},"​")],-1)),s[19]||(s[19]=a("p",null,[t("Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with "),a("code",null,"plot_spaghetti"),t(".")],-1)),a("details",k,[a("summary",null,[s[9]||(s[9]=a("a",{id:"HarmonicBalance.plot_spaghetti",href:"#HarmonicBalance.plot_spaghetti"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_spaghetti")],-1)),s[10]||(s[10]=t()),e(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=n(`
julia
plot_spaghetti(res::Result; x, y, z, kwargs...)
Plot a three dimension line plot of a Result object as a function of the parameters. Works with 1D and 2D datasets.
Class selection done by passing String or Vector{String} as kwarg:
class::String       :   only count solutions in this class ("all" --> plot everything)
-not_class::String   :   do not count solutions in this class
Other kwargs are passed onto Plots.gr()
source
`,6))])])}const C=l(d,[["render",u]]);export{j as __pageData,C as default};
+not_class::String   :   do not count solutions in this class

Other kwargs are passed onto Plots.gr()

source

`,6))]),s[18]||(s[18]=a("h2",{id:"Plot-spaghetti-plot",tabindex:"-1"},[t("Plot spaghetti plot "),a("a",{class:"header-anchor",href:"#Plot-spaghetti-plot","aria-label":'Permalink to "Plot spaghetti plot {#Plot-spaghetti-plot}"'},"​")],-1)),s[19]||(s[19]=a("p",null,[t("Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with "),a("code",null,"plot_spaghetti"),t(".")],-1)),a("details",k,[a("summary",null,[s[9]||(s[9]=a("a",{id:"HarmonicBalance.plot_spaghetti",href:"#HarmonicBalance.plot_spaghetti"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_spaghetti")],-1)),s[10]||(s[10]=t()),n(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e(`
julia
plot_spaghetti(res::Result; x, y, z, kwargs...)

Plot a three dimension line plot of a Result object as a function of the parameters. Works with 1D and 2D datasets.

Class selection done by passing String or Vector{String} as kwarg:

class::String       :   only count solutions in this class ("all" --> plot everything)
+not_class::String   :   do not count solutions in this class

Other kwargs are passed onto Plots.gr()

source

`,6))])])}const C=l(d,[["render",u]]);export{j as __pageData,C as default}; diff --git a/dev/assets/manual_plotting.md.CMWmhlSv.lean.js b/dev/assets/manual_plotting.md.C0lzz6zg.lean.js similarity index 92% rename from dev/assets/manual_plotting.md.CMWmhlSv.lean.js rename to dev/assets/manual_plotting.md.C0lzz6zg.lean.js index d52c02c6..e837e898 100644 --- a/dev/assets/manual_plotting.md.CMWmhlSv.lean.js +++ b/dev/assets/manual_plotting.md.C0lzz6zg.lean.js @@ -1,18 +1,18 @@ -import{_ as l,c as o,j as a,a as t,G as e,a4 as n,B as p,o as r}from"./chunks/framework.DGj8AcR1.js";const j=JSON.parse('{"title":"Analysis and plotting","description":"","frontmatter":{},"headers":[],"relativePath":"manual/plotting.md","filePath":"manual/plotting.md"}'),d={name:"manual/plotting.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""};function u(y,s,m,b,E,f){const i=p("Badge");return r(),o("div",null,[s[12]||(s[12]=a("h1",{id:"Analysis-and-plotting",tabindex:"-1"},[t("Analysis and plotting "),a("a",{class:"header-anchor",href:"#Analysis-and-plotting","aria-label":'Permalink to "Analysis and plotting {#Analysis-and-plotting}"'},"​")],-1)),s[13]||(s[13]=a("p",null,[t("The key method for visualization is "),a("code",null,"transform_solutions"),t(", which parses a string into a symbolic expression and evaluates it for every steady state solution.")],-1)),a("details",h,[a("summary",null,[s[0]||(s[0]=a("a",{id:"HarmonicBalance.transform_solutions",href:"#HarmonicBalance.transform_solutions"},[a("span",{class:"jlbinding"},"HarmonicBalance.transform_solutions")],-1)),s[1]||(s[1]=t()),e(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=n(`
julia
transform_solutions(
+import{_ as l,c as o,j as a,a as t,G as n,a4 as e,B as p,o as r}from"./chunks/framework.DGj8AcR1.js";const j=JSON.parse('{"title":"Analysis and plotting","description":"","frontmatter":{},"headers":[],"relativePath":"manual/plotting.md","filePath":"manual/plotting.md"}'),d={name:"manual/plotting.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""};function u(y,s,m,b,E,f){const i=p("Badge");return r(),o("div",null,[s[12]||(s[12]=a("h1",{id:"Analysis-and-plotting",tabindex:"-1"},[t("Analysis and plotting "),a("a",{class:"header-anchor",href:"#Analysis-and-plotting","aria-label":'Permalink to "Analysis and plotting {#Analysis-and-plotting}"'},"​")],-1)),s[13]||(s[13]=a("p",null,[t("The key method for visualization is "),a("code",null,"transform_solutions"),t(", which parses a string into a symbolic expression and evaluates it for every steady state solution.")],-1)),a("details",h,[a("summary",null,[s[0]||(s[0]=a("a",{id:"HarmonicBalance.transform_solutions",href:"#HarmonicBalance.transform_solutions"},[a("span",{class:"jlbinding"},"HarmonicBalance.transform_solutions")],-1)),s[1]||(s[1]=t()),n(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=e(`
julia
transform_solutions(
     res::HarmonicBalance.Result,
     func;
     branches,
     realify
-) -> Vector
Takes a Result object and a string f representing a Symbolics.jl expression. Returns an array with the values of f evaluated for the respective solutions. Additional substitution rules can be specified in rules in the format ("a" => val) or (a => val)
source
`,3))]),s[14]||(s[14]=a("h2",{id:"Plotting-solutions",tabindex:"-1"},[t("Plotting solutions "),a("a",{class:"header-anchor",href:"#Plotting-solutions","aria-label":'Permalink to "Plotting solutions {#Plotting-solutions}"'},"​")],-1)),s[15]||(s[15]=a("p",null,[t("The function "),a("code",null,"plot"),t(" is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by "),a("code",null,"sort_solutions"),t(".")],-1)),a("details",c,[a("summary",null,[s[3]||(s[3]=a("a",{id:"RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}",href:"#RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}"},[a("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[4]||(s[4]=t()),e(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[5]||(s[5]=n(`
julia
plot(
+) -> Vector
Takes a Result object and a string f representing a Symbolics.jl expression. Returns an array with the values of f evaluated for the respective solutions. Additional substitution rules can be specified in rules in the format ("a" => val) or (a => val)
source
`,3))]),s[14]||(s[14]=a("h2",{id:"Plotting-solutions",tabindex:"-1"},[t("Plotting solutions "),a("a",{class:"header-anchor",href:"#Plotting-solutions","aria-label":'Permalink to "Plotting solutions {#Plotting-solutions}"'},"​")],-1)),s[15]||(s[15]=a("p",null,[t("The function "),a("code",null,"plot"),t(" is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by "),a("code",null,"sort_solutions"),t(".")],-1)),a("details",c,[a("summary",null,[s[3]||(s[3]=a("a",{id:"RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}",href:"#RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}"},[a("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[4]||(s[4]=t()),n(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[5]||(s[5]=e(`
julia
plot(
     res::HarmonicBalance.Result,
     varargs...;
     cut,
     kwargs...
 ) -> Plots.Plot
Plot a Result object.
Class selection done by passing String or Vector{String} as kwarg:
class       :   only plot solutions in this class(es) ("all" --> plot everything)
 not_class   :   do not plot solutions in this class(es)
Other kwargs are passed onto Plots.gr().
See also plot!
The x,y,z arguments are Strings compatible with Symbolics.jl, e.g., y=2*sqrt(u1^2+v1^2) plots the amplitude of the first quadratures multiplied by 2.
1D plots
plot(res::Result; x::String, y::String, class="default", not_class=[], kwargs...)
-plot(res::Result, y::String; kwargs...) # take x automatically from Result
Default behaviour is to plot stable solutions as full lines, unstable as dashed.
If a sweep in two parameters were done, i.e., dim(res)==2, a one dimensional cut can be plotted by using the keyword cut were it takes a Pair{Num, Float64} type entry. For example, plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2)) plots a cut at λ = 0.2.
2D plots
plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)
To make the 2d plot less chaotic it is required to specify the specific branch to plot, labeled by a Int64.
The x and y axes are taken automatically from res
source
`,17))]),s[16]||(s[16]=a("h2",{id:"Plotting-phase-diagrams",tabindex:"-1"},[t("Plotting phase diagrams "),a("a",{class:"header-anchor",href:"#Plotting-phase-diagrams","aria-label":'Permalink to "Plotting phase diagrams {#Plotting-phase-diagrams}"'},"​")],-1)),s[17]||(s[17]=a("p",null,[t("In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. "),a("code",null,"plot_phase_diagram"),t(" handles this for 1D and 2D datasets.")],-1)),a("details",g,[a("summary",null,[s[6]||(s[6]=a("a",{id:"HarmonicBalance.plot_phase_diagram",href:"#HarmonicBalance.plot_phase_diagram"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_phase_diagram")],-1)),s[7]||(s[7]=t()),e(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[8]||(s[8]=n(`
julia
plot_phase_diagram(
+plot(res::Result, y::String; kwargs...) # take x automatically from Result
Default behaviour is to plot stable solutions as full lines, unstable as dashed.
If a sweep in two parameters were done, i.e., dim(res)==2, a one dimensional cut can be plotted by using the keyword cut were it takes a Pair{Num, Float64} type entry. For example, plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2)) plots a cut at λ = 0.2.
2D plots
plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)
To make the 2d plot less chaotic it is required to specify the specific branch to plot, labeled by a Int64.
The x and y axes are taken automatically from res
source
`,17))]),s[16]||(s[16]=a("h2",{id:"Plotting-phase-diagrams",tabindex:"-1"},[t("Plotting phase diagrams "),a("a",{class:"header-anchor",href:"#Plotting-phase-diagrams","aria-label":'Permalink to "Plotting phase diagrams {#Plotting-phase-diagrams}"'},"​")],-1)),s[17]||(s[17]=a("p",null,[t("In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. "),a("code",null,"plot_phase_diagram"),t(" handles this for 1D and 2D datasets.")],-1)),a("details",g,[a("summary",null,[s[6]||(s[6]=a("a",{id:"HarmonicBalance.plot_phase_diagram",href:"#HarmonicBalance.plot_phase_diagram"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_phase_diagram")],-1)),s[7]||(s[7]=t()),n(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[8]||(s[8]=e(`
julia
plot_phase_diagram(
     res::HarmonicBalance.Result;
     kwargs...
 ) -> Plots.Plot
Plot the number of solutions in a Result object as a function of the parameters. Works with 1D and 2D datasets.
Class selection done by passing String or Vector{String} as kwarg:
class::String       :   only count solutions in this class ("all" --> plot everything)
-not_class::String   :   do not count solutions in this class
Other kwargs are passed onto Plots.gr()
source
`,6))]),s[18]||(s[18]=a("h2",{id:"Plot-spaghetti-plot",tabindex:"-1"},[t("Plot spaghetti plot "),a("a",{class:"header-anchor",href:"#Plot-spaghetti-plot","aria-label":'Permalink to "Plot spaghetti plot {#Plot-spaghetti-plot}"'},"​")],-1)),s[19]||(s[19]=a("p",null,[t("Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with "),a("code",null,"plot_spaghetti"),t(".")],-1)),a("details",k,[a("summary",null,[s[9]||(s[9]=a("a",{id:"HarmonicBalance.plot_spaghetti",href:"#HarmonicBalance.plot_spaghetti"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_spaghetti")],-1)),s[10]||(s[10]=t()),e(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=n(`
julia
plot_spaghetti(res::Result; x, y, z, kwargs...)
Plot a three dimension line plot of a Result object as a function of the parameters. Works with 1D and 2D datasets.
Class selection done by passing String or Vector{String} as kwarg:
class::String       :   only count solutions in this class ("all" --> plot everything)
-not_class::String   :   do not count solutions in this class
Other kwargs are passed onto Plots.gr()
source
`,6))])])}const C=l(d,[["render",u]]);export{j as __pageData,C as default};
+not_class::String   :   do not count solutions in this class

Other kwargs are passed onto Plots.gr()

source

`,6))]),s[18]||(s[18]=a("h2",{id:"Plot-spaghetti-plot",tabindex:"-1"},[t("Plot spaghetti plot "),a("a",{class:"header-anchor",href:"#Plot-spaghetti-plot","aria-label":'Permalink to "Plot spaghetti plot {#Plot-spaghetti-plot}"'},"​")],-1)),s[19]||(s[19]=a("p",null,[t("Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with "),a("code",null,"plot_spaghetti"),t(".")],-1)),a("details",k,[a("summary",null,[s[9]||(s[9]=a("a",{id:"HarmonicBalance.plot_spaghetti",href:"#HarmonicBalance.plot_spaghetti"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_spaghetti")],-1)),s[10]||(s[10]=t()),n(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e(`
julia
plot_spaghetti(res::Result; x, y, z, kwargs...)

Plot a three dimension line plot of a Result object as a function of the parameters. Works with 1D and 2D datasets.

Class selection done by passing String or Vector{String} as kwarg:

class::String       :   only count solutions in this class ("all" --> plot everything)
+not_class::String   :   do not count solutions in this class

Other kwargs are passed onto Plots.gr()

source

`,6))])])}const C=l(d,[["render",u]]);export{j as __pageData,C as default}; diff --git a/dev/assets/manual_saving.md.CiRX_Ceq.js b/dev/assets/manual_saving.md.DG3AlMel.js similarity index 66% rename from dev/assets/manual_saving.md.CiRX_Ceq.js rename to dev/assets/manual_saving.md.DG3AlMel.js index 87cf7f7e..aaf2936d 100644 --- a/dev/assets/manual_saving.md.CiRX_Ceq.js +++ b/dev/assets/manual_saving.md.DG3AlMel.js @@ -1 +1 @@ -import{_ as t,c as l,a4 as s,j as a,a as i,G as o,B as c,o as d}from"./chunks/framework.DGj8AcR1.js";const y=JSON.parse('{"title":"Saving and loading","description":"","frontmatter":{},"headers":[],"relativePath":"manual/saving.md","filePath":"manual/saving.md"}'),r={name:"manual/saving.md"},p={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function h(m,e,b,v,f,k){const n=c("Badge");return d(),l("div",null,[e[9]||(e[9]=s('

Saving and loading ​

All of the types native to HarmonicBalance.jl can be saved into a .jld2 file using save and loaded using load. Most of the saving/loading is performed using the package JLD2.jl, with the addition of reinstating the symbolic variables in the HarmonicBalance namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as Result can be saved and loaded with no loss of information.

The function export_csv saves a .csv file which can be plot elsewhere.

',3)),a("details",p,[a("summary",null,[e[0]||(e[0]=a("a",{id:"HarmonicBalance.save",href:"#HarmonicBalance.save"},[a("span",{class:"jlbinding"},"HarmonicBalance.save")],-1)),e[1]||(e[1]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=s('
julia
save(filename, object)

Saves object into .jld2 file filename (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.

source

',3))]),a("details",g,[a("summary",null,[e[3]||(e[3]=a("a",{id:"HarmonicBalance.load",href:"#HarmonicBalance.load"},[a("span",{class:"jlbinding"},"HarmonicBalance.load")],-1)),e[4]||(e[4]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[5]||(e[5]=s('
julia
load(filename)

Loads an object from filename. For objects containing symbolic expressions such as HarmonicEquation, the symbolic variables are reinstated in the HarmonicBalance namespace.

source

',3))]),a("details",u,[a("summary",null,[e[6]||(e[6]=a("a",{id:"HarmonicBalance.export_csv",href:"#HarmonicBalance.export_csv"},[a("span",{class:"jlbinding"},"HarmonicBalance.export_csv")],-1)),e[7]||(e[7]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[8]||(e[8]=s('
julia
export_csv(filename, res, branch)

Saves into filename a specified solution branch of the Result res.

source

',3))]),e[10]||(e[10]=s('
',1))])}const B=t(r,[["render",h]]);export{y as __pageData,B as default}; +import{_ as t,c as l,a4 as s,j as e,a as i,G as o,B as c,o as d}from"./chunks/framework.DGj8AcR1.js";const y=JSON.parse('{"title":"Saving and loading","description":"","frontmatter":{},"headers":[],"relativePath":"manual/saving.md","filePath":"manual/saving.md"}'),r={name:"manual/saving.md"},p={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function h(m,a,b,v,f,k){const n=c("Badge");return d(),l("div",null,[a[9]||(a[9]=s('

Saving and loading ​

All of the types native to HarmonicBalance.jl can be saved into a .jld2 file using save and loaded using load. Most of the saving/loading is performed using the package JLD2.jl, with the addition of reinstating the symbolic variables in the HarmonicBalance namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as Result can be saved and loaded with no loss of information.

The function export_csv saves a .csv file which can be plot elsewhere.

',3)),e("details",p,[e("summary",null,[a[0]||(a[0]=e("a",{id:"HarmonicBalance.save",href:"#HarmonicBalance.save"},[e("span",{class:"jlbinding"},"HarmonicBalance.save")],-1)),a[1]||(a[1]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[2]||(a[2]=s('
julia
save(filename, object)

Saves object into .jld2 file filename (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.

source

',3))]),e("details",g,[e("summary",null,[a[3]||(a[3]=e("a",{id:"HarmonicBalance.load",href:"#HarmonicBalance.load"},[e("span",{class:"jlbinding"},"HarmonicBalance.load")],-1)),a[4]||(a[4]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[5]||(a[5]=s('
julia
load(filename)

Loads an object from filename. For objects containing symbolic expressions such as HarmonicEquation, the symbolic variables are reinstated in the HarmonicBalance namespace.

source

',3))]),e("details",u,[e("summary",null,[a[6]||(a[6]=e("a",{id:"HarmonicBalance.export_csv",href:"#HarmonicBalance.export_csv"},[e("span",{class:"jlbinding"},"HarmonicBalance.export_csv")],-1)),a[7]||(a[7]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[8]||(a[8]=s('
julia
export_csv(filename, res, branch)

Saves into filename a specified solution branch of the Result res.

source

',3))]),a[10]||(a[10]=s('
',1))])}const B=t(r,[["render",h]]);export{y as __pageData,B as default}; diff --git a/dev/assets/manual_saving.md.CiRX_Ceq.lean.js b/dev/assets/manual_saving.md.DG3AlMel.lean.js similarity index 66% rename from dev/assets/manual_saving.md.CiRX_Ceq.lean.js rename to dev/assets/manual_saving.md.DG3AlMel.lean.js index 87cf7f7e..aaf2936d 100644 --- a/dev/assets/manual_saving.md.CiRX_Ceq.lean.js +++ b/dev/assets/manual_saving.md.DG3AlMel.lean.js @@ -1 +1 @@ -import{_ as t,c as l,a4 as s,j as a,a as i,G as o,B as c,o as d}from"./chunks/framework.DGj8AcR1.js";const y=JSON.parse('{"title":"Saving and loading","description":"","frontmatter":{},"headers":[],"relativePath":"manual/saving.md","filePath":"manual/saving.md"}'),r={name:"manual/saving.md"},p={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function h(m,e,b,v,f,k){const n=c("Badge");return d(),l("div",null,[e[9]||(e[9]=s('

Saving and loading ​

All of the types native to HarmonicBalance.jl can be saved into a .jld2 file using save and loaded using load. Most of the saving/loading is performed using the package JLD2.jl, with the addition of reinstating the symbolic variables in the HarmonicBalance namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as Result can be saved and loaded with no loss of information.

The function export_csv saves a .csv file which can be plot elsewhere.

',3)),a("details",p,[a("summary",null,[e[0]||(e[0]=a("a",{id:"HarmonicBalance.save",href:"#HarmonicBalance.save"},[a("span",{class:"jlbinding"},"HarmonicBalance.save")],-1)),e[1]||(e[1]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=s('
julia
save(filename, object)

Saves object into .jld2 file filename (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.

source

',3))]),a("details",g,[a("summary",null,[e[3]||(e[3]=a("a",{id:"HarmonicBalance.load",href:"#HarmonicBalance.load"},[a("span",{class:"jlbinding"},"HarmonicBalance.load")],-1)),e[4]||(e[4]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[5]||(e[5]=s('
julia
load(filename)

Loads an object from filename. For objects containing symbolic expressions such as HarmonicEquation, the symbolic variables are reinstated in the HarmonicBalance namespace.

source

',3))]),a("details",u,[a("summary",null,[e[6]||(e[6]=a("a",{id:"HarmonicBalance.export_csv",href:"#HarmonicBalance.export_csv"},[a("span",{class:"jlbinding"},"HarmonicBalance.export_csv")],-1)),e[7]||(e[7]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[8]||(e[8]=s('
julia
export_csv(filename, res, branch)

Saves into filename a specified solution branch of the Result res.

source

',3))]),e[10]||(e[10]=s('
',1))])}const B=t(r,[["render",h]]);export{y as __pageData,B as default}; +import{_ as t,c as l,a4 as s,j as e,a as i,G as o,B as c,o as d}from"./chunks/framework.DGj8AcR1.js";const y=JSON.parse('{"title":"Saving and loading","description":"","frontmatter":{},"headers":[],"relativePath":"manual/saving.md","filePath":"manual/saving.md"}'),r={name:"manual/saving.md"},p={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function h(m,a,b,v,f,k){const n=c("Badge");return d(),l("div",null,[a[9]||(a[9]=s('

Saving and loading ​

All of the types native to HarmonicBalance.jl can be saved into a .jld2 file using save and loaded using load. Most of the saving/loading is performed using the package JLD2.jl, with the addition of reinstating the symbolic variables in the HarmonicBalance namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as Result can be saved and loaded with no loss of information.

The function export_csv saves a .csv file which can be plot elsewhere.

',3)),e("details",p,[e("summary",null,[a[0]||(a[0]=e("a",{id:"HarmonicBalance.save",href:"#HarmonicBalance.save"},[e("span",{class:"jlbinding"},"HarmonicBalance.save")],-1)),a[1]||(a[1]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[2]||(a[2]=s('
julia
save(filename, object)

Saves object into .jld2 file filename (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.

source

',3))]),e("details",g,[e("summary",null,[a[3]||(a[3]=e("a",{id:"HarmonicBalance.load",href:"#HarmonicBalance.load"},[e("span",{class:"jlbinding"},"HarmonicBalance.load")],-1)),a[4]||(a[4]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[5]||(a[5]=s('
julia
load(filename)

Loads an object from filename. For objects containing symbolic expressions such as HarmonicEquation, the symbolic variables are reinstated in the HarmonicBalance namespace.

source

',3))]),e("details",u,[e("summary",null,[a[6]||(a[6]=e("a",{id:"HarmonicBalance.export_csv",href:"#HarmonicBalance.export_csv"},[e("span",{class:"jlbinding"},"HarmonicBalance.export_csv")],-1)),a[7]||(a[7]=i()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[8]||(a[8]=s('
julia
export_csv(filename, res, branch)

Saves into filename a specified solution branch of the Result res.

source

',3))]),a[10]||(a[10]=s('
',1))])}const B=t(r,[["render",h]]);export{y as __pageData,B as default}; diff --git a/dev/assets/manual_solving_harmonics.md.Cup8_oE9.js b/dev/assets/manual_solving_harmonics.md.Bi1hM2V8.js similarity index 98% rename from dev/assets/manual_solving_harmonics.md.Cup8_oE9.js rename to dev/assets/manual_solving_harmonics.md.Bi1hM2V8.js index ae1c4470..c4ffcfae 100644 --- a/dev/assets/manual_solving_harmonics.md.Cup8_oE9.js +++ b/dev/assets/manual_solving_harmonics.md.Bi1hM2V8.js @@ -1,7 +1,7 @@ import{_ as h,c as l,a4 as t,j as s,a,G as n,B as p,o}from"./chunks/framework.DGj8AcR1.js";const H=JSON.parse('{"title":"Solving harmonic equations","description":"","frontmatter":{},"headers":[],"relativePath":"manual/solving_harmonics.md","filePath":"manual/solving_harmonics.md"}'),r={name:"manual/solving_harmonics.md"},d={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},c={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},g={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.687ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.124ex",height:"2.573ex",role:"img",focusable:"false",viewBox:"0 -833.9 11988.7 1137.4","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},E={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},u={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""};function T(b,i,f,F,C,v){const e=p("Badge");return o(),l("div",null,[i[22]||(i[22]=t('

Solving harmonic equations ​

Once a differential equation of motion has been defined in DifferentialEquation and converted to a HarmonicEquation, we may use the homotopy continuation method (as implemented in HomotopyContinuation.jl) to find steady states. This means that, having called get_harmonic_equations, we need to set all time-derivatives to zero and parse the resulting algebraic equations into a Problem.

Problem holds the steady-state equations, and (optionally) the symbolic Jacobian which is needed for stability / linear response calculations.

Once defined, a Problem can be solved for a set of input parameters using get_steady_states to obtain Result.

',4)),s("details",d,[s("summary",null,[i[0]||(i[0]=s("a",{id:"HarmonicBalance.Problem",href:"#HarmonicBalance.Problem"},[s("span",{class:"jlbinding"},"HarmonicBalance.Problem")],-1)),i[1]||(i[1]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[2]||(i[2]=t(`
julia
mutable struct Problem

Holds a set of algebraic equations describing the steady state of a system.

Fields

  • variables::Vector{Num}: The harmonic variables to be solved for.

  • parameters::Vector{Num}: All symbols which are not the harmonic variables.

  • system::HomotopyContinuation.ModelKit.System: The input object for HomotopyContinuation.jl solver methods.

  • jacobian::Any: The Jacobian matrix (possibly symbolic). If false, the Jacobian is ignored (may be calculated implicitly after solving).

  • eom::HarmonicEquation: The HarmonicEquation object used to generate this Problem.

Constructors

julia
Problem(eom::HarmonicEquation; Jacobian=true) # find and store the symbolic Jacobian
 Problem(eom::HarmonicEquation; Jacobian="implicit") # ignore the Jacobian for now, compute implicitly later
 Problem(eom::HarmonicEquation; Jacobian=J) # use J as the Jacobian (a function that takes a Dict)
-Problem(eom::HarmonicEquation; Jacobian=false) # ignore the Jacobian

source

`,7))]),s("details",k,[s("summary",null,[i[3]||(i[3]=s("a",{id:"HarmonicBalance.get_steady_states",href:"#HarmonicBalance.get_steady_states"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_steady_states")],-1)),i[4]||(i[4]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[11]||(i[11]=t(`
julia
get_steady_states(problem::HarmonicEquation,
+Problem(eom::HarmonicEquation; Jacobian=false) # ignore the Jacobian

source

`,7))]),s("details",k,[s("summary",null,[i[3]||(i[3]=s("a",{id:"HarmonicBalance.get_steady_states",href:"#HarmonicBalance.get_steady_states"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_steady_states")],-1)),i[4]||(i[4]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[11]||(i[11]=t(`
julia
get_steady_states(problem::HarmonicEquation,
                     method::HarmonicBalanceMethod,
                     swept_parameters::ParameterRange,
                     fixed_parameters::ParameterList;
@@ -30,7 +30,7 @@ import{_ as h,c as l,a4 as t,j as s,a,G as n,B as p,o}from"./chunks/framework.DG
        of which real:    1
        of which stable:  1
 
-    Classes: stable, physical, Hopf, binary_labels

source

`,4))]),s("details",u,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.Result",href:"#HarmonicBalance.Result"},[s("span",{class:"jlbinding"},"HarmonicBalance.Result")],-1)),i[14]||(i[14]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[15]||(i[15]=t('
julia
mutable struct Result

Stores the steady states of a HarmonicEquation.

Fields

  • solutions::Array{Vector{Vector{ComplexF64}}}: The variable values of steady-state solutions.

  • swept_parameters::OrderedCollections.OrderedDict{Num, Vector{Union{Float64, ComplexF64}}}: Values of all parameters for all solutions.

  • fixed_parameters::OrderedCollections.OrderedDict{Num, Float64}: The parameters fixed throughout the solutions.

  • problem::HarmonicBalance.Problem: The Problem used to generate this.

  • classes::Dict{String, Array}: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method classify_solutions!).

  • jacobian::Function: The Jacobian with fixed_parameters already substituted. Accepts a dictionary specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function. If problem.jacobian was false, this holds a function that rearranges the equations to find J only after numerical values are inserted (preferable in cases where the symbolic J would be very large).

  • seed::UInt32: Seed used for the solver

source

',5))]),i[23]||(i[23]=s("h2",{id:"Classifying-solutions",tabindex:"-1"},[a("Classifying solutions "),s("a",{class:"header-anchor",href:"#Classifying-solutions","aria-label":'Permalink to "Classifying solutions {#Classifying-solutions}"'},"​")],-1)),i[24]||(i[24]=s("p",null,[a("The solutions in "),s("code",null,"Result"),a(" are accompanied by similarly-sized boolean arrays stored in the dictionary "),s("code",null,"Result.classes"),a(". The classes can be used by the plotting functions to show/hide/label certain solutions.")],-1)),i[25]||(i[25]=s("p",null,[a('By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with '),s("code",null,"classify_solutions!"),a(".")],-1)),s("details",y,[s("summary",null,[i[16]||(i[16]=s("a",{id:"HarmonicBalance.classify_solutions!",href:"#HarmonicBalance.classify_solutions!"},[s("span",{class:"jlbinding"},"HarmonicBalance.classify_solutions!")],-1)),i[17]||(i[17]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[18]||(i[18]=t(`
julia
classify_solutions!(
+    Classes: stable, physical, Hopf, binary_labels

source

`,4))]),s("details",u,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.Result",href:"#HarmonicBalance.Result"},[s("span",{class:"jlbinding"},"HarmonicBalance.Result")],-1)),i[14]||(i[14]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[15]||(i[15]=t('
julia
mutable struct Result

Stores the steady states of a HarmonicEquation.

Fields

  • solutions::Array{Vector{Vector{ComplexF64}}}: The variable values of steady-state solutions.

  • swept_parameters::OrderedCollections.OrderedDict{Num, Vector{Union{Float64, ComplexF64}}}: Values of all parameters for all solutions.

  • fixed_parameters::OrderedCollections.OrderedDict{Num, Float64}: The parameters fixed throughout the solutions.

  • problem::HarmonicBalance.Problem: The Problem used to generate this.

  • classes::Dict{String, Array}: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method classify_solutions!).

  • jacobian::Function: The Jacobian with fixed_parameters already substituted. Accepts a dictionary specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function. If problem.jacobian was false, this holds a function that rearranges the equations to find J only after numerical values are inserted (preferable in cases where the symbolic J would be very large).

  • seed::UInt32: Seed used for the solver

source

',5))]),i[23]||(i[23]=s("h2",{id:"Classifying-solutions",tabindex:"-1"},[a("Classifying solutions "),s("a",{class:"header-anchor",href:"#Classifying-solutions","aria-label":'Permalink to "Classifying solutions {#Classifying-solutions}"'},"​")],-1)),i[24]||(i[24]=s("p",null,[a("The solutions in "),s("code",null,"Result"),a(" are accompanied by similarly-sized boolean arrays stored in the dictionary "),s("code",null,"Result.classes"),a(". The classes can be used by the plotting functions to show/hide/label certain solutions.")],-1)),i[25]||(i[25]=s("p",null,[a('By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with '),s("code",null,"classify_solutions!"),a(".")],-1)),s("details",y,[s("summary",null,[i[16]||(i[16]=s("a",{id:"HarmonicBalance.classify_solutions!",href:"#HarmonicBalance.classify_solutions!"},[s("span",{class:"jlbinding"},"HarmonicBalance.classify_solutions!")],-1)),i[17]||(i[17]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[18]||(i[18]=t(`
julia
classify_solutions!(
     res::HarmonicBalance.Result,
     func::Union{Function, String},
     name::String;
@@ -39,8 +39,8 @@ import{_ as h,c as l,a4 as t,j as s,a,G as n,B as p,o}from"./chunks/framework.DG
 res = get_steady_states(problem, swept_parameters, fixed_parameters)
 
 # classify, store in result.classes["large_amplitude"]
-classify_solutions!(res, "sqrt(u1^2 + v1^2) > 1.0" , "large_amplitude")

source

`,7))]),i[26]||(i[26]=t('

Sorting solutions ​

Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.

For stable states, the branches describe a system's behaviour under adiabatic parameter changes.

Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.

The function sort_solutions goes over the the raw output of get_steady_states and sorts each entry such that neighboring solution sets minimize Euclidean distance.

Currently, sort_solutions is compatible with 1D and 2D arrays of solution sets.

',6)),s("details",Q,[s("summary",null,[i[19]||(i[19]=s("a",{id:"HarmonicBalance.sort_solutions",href:"#HarmonicBalance.sort_solutions"},[s("span",{class:"jlbinding"},"HarmonicBalance.sort_solutions")],-1)),i[20]||(i[20]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[21]||(i[21]=t(`
julia
sort_solutions(
+classify_solutions!(res, "sqrt(u1^2 + v1^2) > 1.0" , "large_amplitude")

source

`,7))]),i[26]||(i[26]=t('

Sorting solutions ​

Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.

For stable states, the branches describe a system's behaviour under adiabatic parameter changes.

Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.

The function sort_solutions goes over the the raw output of get_steady_states and sorts each entry such that neighboring solution sets minimize Euclidean distance.

Currently, sort_solutions is compatible with 1D and 2D arrays of solution sets.

',6)),s("details",Q,[s("summary",null,[i[19]||(i[19]=s("a",{id:"HarmonicBalance.sort_solutions",href:"#HarmonicBalance.sort_solutions"},[s("span",{class:"jlbinding"},"HarmonicBalance.sort_solutions")],-1)),i[20]||(i[20]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[21]||(i[21]=t(`
julia
sort_solutions(
     solutions::Array;
     sorting,
     show_progress
-) -> Array

Sorts solutions into branches according to the method sorting.

solutions is an n-dimensional array of Vector{Vector}. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.

Keyword arguments

  • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".

  • show_progress: Indicate whether a progress bar should be displayed.

source

`,6))])])}const w=h(r,[["render",T]]);export{H as __pageData,w as default}; +) -> Array

Sorts solutions into branches according to the method sorting.

solutions is an n-dimensional array of Vector{Vector}. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.

Keyword arguments

  • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".

  • show_progress: Indicate whether a progress bar should be displayed.

source

`,6))])])}const w=h(r,[["render",T]]);export{H as __pageData,w as default}; diff --git a/dev/assets/manual_solving_harmonics.md.Cup8_oE9.lean.js b/dev/assets/manual_solving_harmonics.md.Bi1hM2V8.lean.js similarity index 98% rename from dev/assets/manual_solving_harmonics.md.Cup8_oE9.lean.js rename to dev/assets/manual_solving_harmonics.md.Bi1hM2V8.lean.js index ae1c4470..c4ffcfae 100644 --- a/dev/assets/manual_solving_harmonics.md.Cup8_oE9.lean.js +++ b/dev/assets/manual_solving_harmonics.md.Bi1hM2V8.lean.js @@ -1,7 +1,7 @@ import{_ as h,c as l,a4 as t,j as s,a,G as n,B as p,o}from"./chunks/framework.DGj8AcR1.js";const H=JSON.parse('{"title":"Solving harmonic equations","description":"","frontmatter":{},"headers":[],"relativePath":"manual/solving_harmonics.md","filePath":"manual/solving_harmonics.md"}'),r={name:"manual/solving_harmonics.md"},d={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},c={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},g={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.687ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.124ex",height:"2.573ex",role:"img",focusable:"false",viewBox:"0 -833.9 11988.7 1137.4","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},E={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},u={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""};function T(b,i,f,F,C,v){const e=p("Badge");return o(),l("div",null,[i[22]||(i[22]=t('

Solving harmonic equations ​

Once a differential equation of motion has been defined in DifferentialEquation and converted to a HarmonicEquation, we may use the homotopy continuation method (as implemented in HomotopyContinuation.jl) to find steady states. This means that, having called get_harmonic_equations, we need to set all time-derivatives to zero and parse the resulting algebraic equations into a Problem.

Problem holds the steady-state equations, and (optionally) the symbolic Jacobian which is needed for stability / linear response calculations.

Once defined, a Problem can be solved for a set of input parameters using get_steady_states to obtain Result.

',4)),s("details",d,[s("summary",null,[i[0]||(i[0]=s("a",{id:"HarmonicBalance.Problem",href:"#HarmonicBalance.Problem"},[s("span",{class:"jlbinding"},"HarmonicBalance.Problem")],-1)),i[1]||(i[1]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[2]||(i[2]=t(`
julia
mutable struct Problem

Holds a set of algebraic equations describing the steady state of a system.

Fields

  • variables::Vector{Num}: The harmonic variables to be solved for.

  • parameters::Vector{Num}: All symbols which are not the harmonic variables.

  • system::HomotopyContinuation.ModelKit.System: The input object for HomotopyContinuation.jl solver methods.

  • jacobian::Any: The Jacobian matrix (possibly symbolic). If false, the Jacobian is ignored (may be calculated implicitly after solving).

  • eom::HarmonicEquation: The HarmonicEquation object used to generate this Problem.

Constructors

julia
Problem(eom::HarmonicEquation; Jacobian=true) # find and store the symbolic Jacobian
 Problem(eom::HarmonicEquation; Jacobian="implicit") # ignore the Jacobian for now, compute implicitly later
 Problem(eom::HarmonicEquation; Jacobian=J) # use J as the Jacobian (a function that takes a Dict)
-Problem(eom::HarmonicEquation; Jacobian=false) # ignore the Jacobian

source

`,7))]),s("details",k,[s("summary",null,[i[3]||(i[3]=s("a",{id:"HarmonicBalance.get_steady_states",href:"#HarmonicBalance.get_steady_states"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_steady_states")],-1)),i[4]||(i[4]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[11]||(i[11]=t(`
julia
get_steady_states(problem::HarmonicEquation,
+Problem(eom::HarmonicEquation; Jacobian=false) # ignore the Jacobian

source

`,7))]),s("details",k,[s("summary",null,[i[3]||(i[3]=s("a",{id:"HarmonicBalance.get_steady_states",href:"#HarmonicBalance.get_steady_states"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_steady_states")],-1)),i[4]||(i[4]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[11]||(i[11]=t(`
julia
get_steady_states(problem::HarmonicEquation,
                     method::HarmonicBalanceMethod,
                     swept_parameters::ParameterRange,
                     fixed_parameters::ParameterList;
@@ -30,7 +30,7 @@ import{_ as h,c as l,a4 as t,j as s,a,G as n,B as p,o}from"./chunks/framework.DG
        of which real:    1
        of which stable:  1
 
-    Classes: stable, physical, Hopf, binary_labels

source

`,4))]),s("details",u,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.Result",href:"#HarmonicBalance.Result"},[s("span",{class:"jlbinding"},"HarmonicBalance.Result")],-1)),i[14]||(i[14]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[15]||(i[15]=t('
julia
mutable struct Result

Stores the steady states of a HarmonicEquation.

Fields

  • solutions::Array{Vector{Vector{ComplexF64}}}: The variable values of steady-state solutions.

  • swept_parameters::OrderedCollections.OrderedDict{Num, Vector{Union{Float64, ComplexF64}}}: Values of all parameters for all solutions.

  • fixed_parameters::OrderedCollections.OrderedDict{Num, Float64}: The parameters fixed throughout the solutions.

  • problem::HarmonicBalance.Problem: The Problem used to generate this.

  • classes::Dict{String, Array}: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method classify_solutions!).

  • jacobian::Function: The Jacobian with fixed_parameters already substituted. Accepts a dictionary specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function. If problem.jacobian was false, this holds a function that rearranges the equations to find J only after numerical values are inserted (preferable in cases where the symbolic J would be very large).

  • seed::UInt32: Seed used for the solver

source

',5))]),i[23]||(i[23]=s("h2",{id:"Classifying-solutions",tabindex:"-1"},[a("Classifying solutions "),s("a",{class:"header-anchor",href:"#Classifying-solutions","aria-label":'Permalink to "Classifying solutions {#Classifying-solutions}"'},"​")],-1)),i[24]||(i[24]=s("p",null,[a("The solutions in "),s("code",null,"Result"),a(" are accompanied by similarly-sized boolean arrays stored in the dictionary "),s("code",null,"Result.classes"),a(". The classes can be used by the plotting functions to show/hide/label certain solutions.")],-1)),i[25]||(i[25]=s("p",null,[a('By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with '),s("code",null,"classify_solutions!"),a(".")],-1)),s("details",y,[s("summary",null,[i[16]||(i[16]=s("a",{id:"HarmonicBalance.classify_solutions!",href:"#HarmonicBalance.classify_solutions!"},[s("span",{class:"jlbinding"},"HarmonicBalance.classify_solutions!")],-1)),i[17]||(i[17]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[18]||(i[18]=t(`
julia
classify_solutions!(
+    Classes: stable, physical, Hopf, binary_labels

source

`,4))]),s("details",u,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.Result",href:"#HarmonicBalance.Result"},[s("span",{class:"jlbinding"},"HarmonicBalance.Result")],-1)),i[14]||(i[14]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[15]||(i[15]=t('
julia
mutable struct Result

Stores the steady states of a HarmonicEquation.

Fields

  • solutions::Array{Vector{Vector{ComplexF64}}}: The variable values of steady-state solutions.

  • swept_parameters::OrderedCollections.OrderedDict{Num, Vector{Union{Float64, ComplexF64}}}: Values of all parameters for all solutions.

  • fixed_parameters::OrderedCollections.OrderedDict{Num, Float64}: The parameters fixed throughout the solutions.

  • problem::HarmonicBalance.Problem: The Problem used to generate this.

  • classes::Dict{String, Array}: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method classify_solutions!).

  • jacobian::Function: The Jacobian with fixed_parameters already substituted. Accepts a dictionary specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function. If problem.jacobian was false, this holds a function that rearranges the equations to find J only after numerical values are inserted (preferable in cases where the symbolic J would be very large).

  • seed::UInt32: Seed used for the solver

source

',5))]),i[23]||(i[23]=s("h2",{id:"Classifying-solutions",tabindex:"-1"},[a("Classifying solutions "),s("a",{class:"header-anchor",href:"#Classifying-solutions","aria-label":'Permalink to "Classifying solutions {#Classifying-solutions}"'},"​")],-1)),i[24]||(i[24]=s("p",null,[a("The solutions in "),s("code",null,"Result"),a(" are accompanied by similarly-sized boolean arrays stored in the dictionary "),s("code",null,"Result.classes"),a(". The classes can be used by the plotting functions to show/hide/label certain solutions.")],-1)),i[25]||(i[25]=s("p",null,[a('By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with '),s("code",null,"classify_solutions!"),a(".")],-1)),s("details",y,[s("summary",null,[i[16]||(i[16]=s("a",{id:"HarmonicBalance.classify_solutions!",href:"#HarmonicBalance.classify_solutions!"},[s("span",{class:"jlbinding"},"HarmonicBalance.classify_solutions!")],-1)),i[17]||(i[17]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[18]||(i[18]=t(`
julia
classify_solutions!(
     res::HarmonicBalance.Result,
     func::Union{Function, String},
     name::String;
@@ -39,8 +39,8 @@ import{_ as h,c as l,a4 as t,j as s,a,G as n,B as p,o}from"./chunks/framework.DG
 res = get_steady_states(problem, swept_parameters, fixed_parameters)
 
 # classify, store in result.classes["large_amplitude"]
-classify_solutions!(res, "sqrt(u1^2 + v1^2) > 1.0" , "large_amplitude")

source

`,7))]),i[26]||(i[26]=t('

Sorting solutions ​

Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.

For stable states, the branches describe a system's behaviour under adiabatic parameter changes.

Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.

The function sort_solutions goes over the the raw output of get_steady_states and sorts each entry such that neighboring solution sets minimize Euclidean distance.

Currently, sort_solutions is compatible with 1D and 2D arrays of solution sets.

',6)),s("details",Q,[s("summary",null,[i[19]||(i[19]=s("a",{id:"HarmonicBalance.sort_solutions",href:"#HarmonicBalance.sort_solutions"},[s("span",{class:"jlbinding"},"HarmonicBalance.sort_solutions")],-1)),i[20]||(i[20]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[21]||(i[21]=t(`
julia
sort_solutions(
+classify_solutions!(res, "sqrt(u1^2 + v1^2) > 1.0" , "large_amplitude")

source

`,7))]),i[26]||(i[26]=t('

Sorting solutions ​

Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.

For stable states, the branches describe a system's behaviour under adiabatic parameter changes.

Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.

The function sort_solutions goes over the the raw output of get_steady_states and sorts each entry such that neighboring solution sets minimize Euclidean distance.

Currently, sort_solutions is compatible with 1D and 2D arrays of solution sets.

',6)),s("details",Q,[s("summary",null,[i[19]||(i[19]=s("a",{id:"HarmonicBalance.sort_solutions",href:"#HarmonicBalance.sort_solutions"},[s("span",{class:"jlbinding"},"HarmonicBalance.sort_solutions")],-1)),i[20]||(i[20]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[21]||(i[21]=t(`
julia
sort_solutions(
     solutions::Array;
     sorting,
     show_progress
-) -> Array

Sorts solutions into branches according to the method sorting.

solutions is an n-dimensional array of Vector{Vector}. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.

Keyword arguments

  • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".

  • show_progress: Indicate whether a progress bar should be displayed.

source

`,6))])])}const w=h(r,[["render",T]]);export{H as __pageData,w as default}; +) -> Array

Sorts solutions into branches according to the method sorting.

solutions is an n-dimensional array of Vector{Vector}. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.

Keyword arguments

  • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".

  • show_progress: Indicate whether a progress bar should be displayed.

source

`,6))])])}const w=h(r,[["render",T]]);export{H as __pageData,w as default}; diff --git a/dev/assets/manual_time_dependent.md.BeG_NgwS.js b/dev/assets/manual_time_dependent.md.uGO7eGa5.js similarity index 97% rename from dev/assets/manual_time_dependent.md.BeG_NgwS.js rename to dev/assets/manual_time_dependent.md.uGO7eGa5.js index 397bdd36..24f8a524 100644 --- a/dev/assets/manual_time_dependent.md.BeG_NgwS.js +++ b/dev/assets/manual_time_dependent.md.uGO7eGa5.js @@ -4,7 +4,7 @@ import{_ as l,c as p,a4 as e,j as i,a,G as t,B as h,o as k}from"./chunks/framewo u0::Vector, sweep::AdiabaticSweep, timespan::Tuple - )

Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in eom to simulate time-evolution within timespan. fixed_parameters must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If u0 is specified, it is used as an initial condition; otherwise the values from fixed_parameters are used.

source

`,3))]),i("details",d,[i("summary",null,[s[3]||(s[3]=i("a",{id:"HarmonicBalance.AdiabaticSweep",href:"#HarmonicBalance.AdiabaticSweep"},[i("span",{class:"jlbinding"},"HarmonicBalance.AdiabaticSweep")],-1)),s[4]||(s[4]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=e(`

Represents a sweep of one or more parameters of a HarmonicEquation. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.

Sweeps of different variables can be combined using +.

Fields

  • functions::Dict{Num, Function}: Maps each swept parameter to a function.

Examples

julia
# create a sweep of parameter a from 0 to 1 over time 0 -> 100
+        )

Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in eom to simulate time-evolution within timespan. fixed_parameters must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If u0 is specified, it is used as an initial condition; otherwise the values from fixed_parameters are used.

source

`,3))]),i("details",d,[i("summary",null,[s[3]||(s[3]=i("a",{id:"HarmonicBalance.AdiabaticSweep",href:"#HarmonicBalance.AdiabaticSweep"},[i("span",{class:"jlbinding"},"HarmonicBalance.AdiabaticSweep")],-1)),s[4]||(s[4]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=e(`

Represents a sweep of one or more parameters of a HarmonicEquation. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.

Sweeps of different variables can be combined using +.

Fields

  • functions::Dict{Num, Function}: Maps each swept parameter to a function.

Examples

julia
# create a sweep of parameter a from 0 to 1 over time 0 -> 100
 julia> @variables a,b;
 julia> sweep = AdiabaticSweep(a => [0., 1.], (0, 100));
 julia> sweep[a](50)
@@ -16,14 +16,14 @@ import{_ as l,c as p,a4 as e,j as i,a,G as t,B as h,o as k}from"./chunks/framewo
 julia> sweep = AdiabaticSweep([a => [0.,1.], b => [0., 1.]], (0,100))

Successive sweeps can be combined,

julia
sweep1 = AdiabaticSweep(ω => [0.95, 1.0], (0, 2e4))
 sweep2 = AdiabaticSweep(λ => [0.05, 0.01], (2e4, 4e4))
 sweep = sweep1 + sweep2

multiple parameters can be swept simultaneously,

julia
sweep = AdiabaticSweep([ω => [0.95;1.0], λ => [5e-2;1e-2]], (0, 2e4))

and custom sweep functions may be used.

julia
ωfunc(t) = cos(t)
-sweep = AdiabaticSweep(ω => ωfunc)

source

`,13))]),s[13]||(s[13]=i("h2",{id:"plotting",tabindex:"-1"},[a("Plotting "),i("a",{class:"header-anchor",href:"#plotting","aria-label":'Permalink to "Plotting"'},"​")],-1)),i("details",E,[i("summary",null,[s[6]||(s[6]=i("a",{id:"RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}",href:"#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[7]||(s[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[8]||(s[8]=e('
julia
plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

Plot a function f of a time-dependent solution soln of harm_eq.

As a function of time

plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

f is parsed by Symbolics.jl

parametric plots

plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)

Parametric plot of f[1] against f[2]

Also callable as plot!

source

',10))]),s[14]||(s[14]=i("h2",{id:"miscellaneous",tabindex:"-1"},[a("Miscellaneous "),i("a",{class:"header-anchor",href:"#miscellaneous","aria-label":'Permalink to "Miscellaneous"'},"​")],-1)),s[15]||(s[15]=i("p",null,"Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.",-1)),i("details",c,[i("summary",null,[s[9]||(s[9]=i("a",{id:"HarmonicBalance.is_stable",href:"#HarmonicBalance.is_stable"},[i("span",{class:"jlbinding"},"HarmonicBalance.is_stable")],-1)),s[10]||(s[10]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e(`
julia
is_stable(
+sweep = AdiabaticSweep(ω => ωfunc)

source

`,13))]),s[13]||(s[13]=i("h2",{id:"plotting",tabindex:"-1"},[a("Plotting "),i("a",{class:"header-anchor",href:"#plotting","aria-label":'Permalink to "Plotting"'},"​")],-1)),i("details",E,[i("summary",null,[s[6]||(s[6]=i("a",{id:"RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}",href:"#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[7]||(s[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[8]||(s[8]=e('
julia
plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

Plot a function f of a time-dependent solution soln of harm_eq.

As a function of time

plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

f is parsed by Symbolics.jl

parametric plots

plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)

Parametric plot of f[1] against f[2]

Also callable as plot!

source

',10))]),s[14]||(s[14]=i("h2",{id:"miscellaneous",tabindex:"-1"},[a("Miscellaneous "),i("a",{class:"header-anchor",href:"#miscellaneous","aria-label":'Permalink to "Miscellaneous"'},"​")],-1)),s[15]||(s[15]=i("p",null,"Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.",-1)),i("details",c,[i("summary",null,[s[9]||(s[9]=i("a",{id:"HarmonicBalance.is_stable",href:"#HarmonicBalance.is_stable"},[i("span",{class:"jlbinding"},"HarmonicBalance.is_stable")],-1)),s[10]||(s[10]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e(`
julia
is_stable(
     soln::OrderedCollections.OrderedDict{Num, ComplexF64},
     eom::HarmonicEquation;
     timespan,
     tol,
     perturb_initial
-)

Numerically investigate the stability of a solution soln of eom within timespan. The initial condition is displaced by perturb_initial.

Return true the solution evolves within tol of the initial value (interpreted as stable).

source

julia
is_stable(
+)

Numerically investigate the stability of a solution soln of eom within timespan. The initial condition is displaced by perturb_initial.

Return true the solution evolves within tol of the initial value (interpreted as stable).

source

julia
is_stable(
     soln::OrderedCollections.OrderedDict{Num, ComplexF64},
     res::HarmonicBalance.Result;
     kwargs...
-) -> Any

Returns true if the solution soln of the Result res is stable. Stable solutions are real and have all Jacobian eigenvalues Re[λ] <= 0. im_tol : an absolute threshold to distinguish real/complex numbers. rel_tol: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)

source

`,7))])])}const v=l(o,[["render",g]]);export{f as __pageData,v as default}; +) -> Any

Returns true if the solution soln of the Result res is stable. Stable solutions are real and have all Jacobian eigenvalues Re[λ] <= 0. im_tol : an absolute threshold to distinguish real/complex numbers. rel_tol: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)

source

`,7))])])}const v=l(o,[["render",g]]);export{f as __pageData,v as default}; diff --git a/dev/assets/manual_time_dependent.md.BeG_NgwS.lean.js b/dev/assets/manual_time_dependent.md.uGO7eGa5.lean.js similarity index 97% rename from dev/assets/manual_time_dependent.md.BeG_NgwS.lean.js rename to dev/assets/manual_time_dependent.md.uGO7eGa5.lean.js index 397bdd36..24f8a524 100644 --- a/dev/assets/manual_time_dependent.md.BeG_NgwS.lean.js +++ b/dev/assets/manual_time_dependent.md.uGO7eGa5.lean.js @@ -4,7 +4,7 @@ import{_ as l,c as p,a4 as e,j as i,a,G as t,B as h,o as k}from"./chunks/framewo u0::Vector, sweep::AdiabaticSweep, timespan::Tuple - )

Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in eom to simulate time-evolution within timespan. fixed_parameters must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If u0 is specified, it is used as an initial condition; otherwise the values from fixed_parameters are used.

source

`,3))]),i("details",d,[i("summary",null,[s[3]||(s[3]=i("a",{id:"HarmonicBalance.AdiabaticSweep",href:"#HarmonicBalance.AdiabaticSweep"},[i("span",{class:"jlbinding"},"HarmonicBalance.AdiabaticSweep")],-1)),s[4]||(s[4]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=e(`

Represents a sweep of one or more parameters of a HarmonicEquation. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.

Sweeps of different variables can be combined using +.

Fields

  • functions::Dict{Num, Function}: Maps each swept parameter to a function.

Examples

julia
# create a sweep of parameter a from 0 to 1 over time 0 -> 100
+        )

Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in eom to simulate time-evolution within timespan. fixed_parameters must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If u0 is specified, it is used as an initial condition; otherwise the values from fixed_parameters are used.

source

`,3))]),i("details",d,[i("summary",null,[s[3]||(s[3]=i("a",{id:"HarmonicBalance.AdiabaticSweep",href:"#HarmonicBalance.AdiabaticSweep"},[i("span",{class:"jlbinding"},"HarmonicBalance.AdiabaticSweep")],-1)),s[4]||(s[4]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=e(`

Represents a sweep of one or more parameters of a HarmonicEquation. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.

Sweeps of different variables can be combined using +.

Fields

  • functions::Dict{Num, Function}: Maps each swept parameter to a function.

Examples

julia
# create a sweep of parameter a from 0 to 1 over time 0 -> 100
 julia> @variables a,b;
 julia> sweep = AdiabaticSweep(a => [0., 1.], (0, 100));
 julia> sweep[a](50)
@@ -16,14 +16,14 @@ import{_ as l,c as p,a4 as e,j as i,a,G as t,B as h,o as k}from"./chunks/framewo
 julia> sweep = AdiabaticSweep([a => [0.,1.], b => [0., 1.]], (0,100))

Successive sweeps can be combined,

julia
sweep1 = AdiabaticSweep(ω => [0.95, 1.0], (0, 2e4))
 sweep2 = AdiabaticSweep(λ => [0.05, 0.01], (2e4, 4e4))
 sweep = sweep1 + sweep2

multiple parameters can be swept simultaneously,

julia
sweep = AdiabaticSweep([ω => [0.95;1.0], λ => [5e-2;1e-2]], (0, 2e4))

and custom sweep functions may be used.

julia
ωfunc(t) = cos(t)
-sweep = AdiabaticSweep(ω => ωfunc)

source

`,13))]),s[13]||(s[13]=i("h2",{id:"plotting",tabindex:"-1"},[a("Plotting "),i("a",{class:"header-anchor",href:"#plotting","aria-label":'Permalink to "Plotting"'},"​")],-1)),i("details",E,[i("summary",null,[s[6]||(s[6]=i("a",{id:"RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}",href:"#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[7]||(s[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[8]||(s[8]=e('
julia
plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

Plot a function f of a time-dependent solution soln of harm_eq.

As a function of time

plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

f is parsed by Symbolics.jl

parametric plots

plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)

Parametric plot of f[1] against f[2]

Also callable as plot!

source

',10))]),s[14]||(s[14]=i("h2",{id:"miscellaneous",tabindex:"-1"},[a("Miscellaneous "),i("a",{class:"header-anchor",href:"#miscellaneous","aria-label":'Permalink to "Miscellaneous"'},"​")],-1)),s[15]||(s[15]=i("p",null,"Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.",-1)),i("details",c,[i("summary",null,[s[9]||(s[9]=i("a",{id:"HarmonicBalance.is_stable",href:"#HarmonicBalance.is_stable"},[i("span",{class:"jlbinding"},"HarmonicBalance.is_stable")],-1)),s[10]||(s[10]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e(`
julia
is_stable(
+sweep = AdiabaticSweep(ω => ωfunc)

source

`,13))]),s[13]||(s[13]=i("h2",{id:"plotting",tabindex:"-1"},[a("Plotting "),i("a",{class:"header-anchor",href:"#plotting","aria-label":'Permalink to "Plotting"'},"​")],-1)),i("details",E,[i("summary",null,[s[6]||(s[6]=i("a",{id:"RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}",href:"#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[7]||(s[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[8]||(s[8]=e('
julia
plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

Plot a function f of a time-dependent solution soln of harm_eq.

As a function of time

plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

f is parsed by Symbolics.jl

parametric plots

plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)

Parametric plot of f[1] against f[2]

Also callable as plot!

source

',10))]),s[14]||(s[14]=i("h2",{id:"miscellaneous",tabindex:"-1"},[a("Miscellaneous "),i("a",{class:"header-anchor",href:"#miscellaneous","aria-label":'Permalink to "Miscellaneous"'},"​")],-1)),s[15]||(s[15]=i("p",null,"Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.",-1)),i("details",c,[i("summary",null,[s[9]||(s[9]=i("a",{id:"HarmonicBalance.is_stable",href:"#HarmonicBalance.is_stable"},[i("span",{class:"jlbinding"},"HarmonicBalance.is_stable")],-1)),s[10]||(s[10]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e(`
julia
is_stable(
     soln::OrderedCollections.OrderedDict{Num, ComplexF64},
     eom::HarmonicEquation;
     timespan,
     tol,
     perturb_initial
-)

Numerically investigate the stability of a solution soln of eom within timespan. The initial condition is displaced by perturb_initial.

Return true the solution evolves within tol of the initial value (interpreted as stable).

source

julia
is_stable(
+)

Numerically investigate the stability of a solution soln of eom within timespan. The initial condition is displaced by perturb_initial.

Return true the solution evolves within tol of the initial value (interpreted as stable).

source

julia
is_stable(
     soln::OrderedCollections.OrderedDict{Num, ComplexF64},
     res::HarmonicBalance.Result;
     kwargs...
-) -> Any

Returns true if the solution soln of the Result res is stable. Stable solutions are real and have all Jacobian eigenvalues Re[λ] <= 0. im_tol : an absolute threshold to distinguish real/complex numbers. rel_tol: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)

source

`,7))])])}const v=l(o,[["render",g]]);export{f as __pageData,v as default}; +) -> Any

Returns true if the solution soln of the Result res is stable. Stable solutions are real and have all Jacobian eigenvalues Re[λ] <= 0. im_tol : an absolute threshold to distinguish real/complex numbers. rel_tol: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)

source

`,7))])])}const v=l(o,[["render",g]]);export{f as __pageData,v as default}; diff --git a/dev/assets/style.CSaE5U6b.css b/dev/assets/style.CSaE5U6b.css new file mode 100644 index 00000000..c68f8282 --- /dev/null +++ b/dev/assets/style.CSaE5U6b.css @@ -0,0 +1 @@ +.img-box[data-v-7654366a]{box-sizing:content-box;border-radius:14px;margin:10px;height:190px;width:140px;overflow:hidden;display:inline-block;color:#fff;position:relative;background-color:transparent;border:2px solid var(--vp-c-bg-alt)}.img-box img[data-v-7654366a]{height:100%;width:100%;object-fit:cover;opacity:.4;transition:transform .3s ease,opacity .3s ease}.caption[data-v-7654366a]{position:absolute;bottom:30px;color:var(--vp-c-text-1);left:10px;opacity:1;transition:transform .3s ease,opacity .3s ease}.subcaption[data-v-7654366a]{position:absolute;bottom:5px;color:var(--vp-c-text-1);left:10px;opacity:0;transition:transform .3s ease,opacity .3s ease}.subcaption 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HarmonicBalance.jl

Appearance

The method of harmonic balance ​

Frequency conversion in oscillating nonlinear systems ​

HarmonicBalance.jl focuses on harmonically-driven nonlinear systems, i.e., dynamical systems governed by equations of motion where all explicitly time-dependent terms are harmonic. Let us take a general nonlinear system of N second-order ODEs with real variables xi(t), i=1,2,⋯,N and time t as the independent variable,

x¨(t)+F(x(t),t)=0.

The vector x(t)=(x1(t),...,xN(t))T fully describes the state of the system. Physically, x(t) encompasses the amplitudes of either point-like or collective oscillators (e.g., mechanical resonators, voltage oscillations in RLC circuits, an oscillating electrical dipole moment, or standing modes of an optical cavity).

As the simplest example, let us first solve the harmonic oscillator in frequency space. The equation of motion is

x¨(t)+γx˙(t)+ω02x(t)=Fcos⁡(ωdt)

where γ is the damping coefficient and ω0 the natural frequency. Fourier-transforming both sides of this equation gives

(ω02−ω2+iωγ)x~(ω)=F2[δ(ω+ωd)+δ(ω−ωd)].

Evidently, x~(ω) is only nonvanishing for ω=±ωd. The system thus responds at the driving frequency only - the behaviour can be captured by a single harmonic. This illustrates the general point that linear systems are exactly solvable by transforming to Fourier space, where the equations are diagonal.

The situation becomes more complex if nonlinear terms are present, as these cause frequency conversion. Suppose we add a quadratic nonlinearity βx2(t) to the equations of motion; an attempt to Fourier-transform gives

FT[x2](ω)=∫x2(t)e−iωtdt=∫−∞+∞x~(ω′)x~(ω″)δ(ω″+ω′−ω)dω′dω″,

which couples all harmonics ω,ω′,ω″ such that ω+ω′+ω″=0. To lowest order, this means the induced motion at the drive frequency generates a higher harmonic, ωd→2ωd. To higher orders however, the frequency conversion propagates through the spectrum, coupling an infinite number of harmonics. The system is not solvable in Fourier space anymore!

Harmonic ansatz & harmonic equations ​

Even though we need an infinity of Fourier components to describe our system exactly, some components are more important than others. The strategy of harmonic balance is to describe the motion of any variable xi(t) in a truncated Fourier space

xi(t)=∑j=1Miui,j(T)cos⁡(ωi,jt)+vi,j(T)sin⁡(ωi,jt).

Within this space, the system is described by a finite-dimensional vector

u(T)=(u1,1(T),v1,1(T),…uN,MN(T),vN,MN(T))

Under the assumption that u(T) evolves at much slower timescales than the oscillatory terms ωi,jt, we may neglect all of its higher order time derivatives. Notice that once ansatz \eqref{eq:harmansatz} is used in Eq. \eqref{eq:ode}, all terms become oscillatory - each prefactor of cos⁡(ωi,jt) and sin⁡(ωi,jt) thus generates a separate equation. Collecting these, we obtain a 1st order nonlinear ODEs,

du(T)dT=F¯(u),

which we call the harmonic equations. The main purpose of HarmonicBalance.jl is to obtain and solve them. We are primarily interested in steady states u0 defined by F¯(u0)=0.

The process of obtaining the harmonic equations is best shown on an example.

Example: the Duffing oscillator ​

Here, we derive the harmonic equations for a single Duffing resonator, governed by the equation

x¨(t)+ω02x(t)+αx3(t)=Fcos⁡(ωdt+θ).

As explained in above, for a periodic driving at frequency ωd and a weak nonlinearity α, we expect the response at frequency ωd to dominate, followed by a response at 3ωd due to frequency conversion.

Single-frequency ansatz ​

We first attempt to describe the steady states of Eq. \eqref{eq:duffing} using only one harmonic, ωd. The starting point is the harmonic ansatz for x

x(t)=u(T)cos⁡(ωdt)+v(T)sin⁡(ωdt),

with the harmonic variables u and v. The slow time T is, for now, equivalent to t. Substituting this ansatz into mechanical equations of motion results in

[u¨+2ωdv˙+u(ω02−ωd2)+3α(u3+uv2)4+Fcos⁡θ]cos⁡(ωdt)+[v¨−2ωdu˙+v(ω02−ωd2)+3α(v3+u2v)4−Fsin⁡θ]sin⁡(ωdt)+α(u3−3uv2)4cos⁡(3ωdt)+α(3u2v−v3)4sin⁡(3ωdt)=0.

We see that the x3 term has generated terms that oscillate at 3ωd, describing the process of frequency upconversion. We now Fourier-transform both sides of Eq. \eqref{eq:ansatz1} with respect to ωd to obtain the harmonic equations. This process is equivalent to extracting the respective coefficients of cos⁡(ωdt) and sin⁡(ωdt). Here the distinction between t and T becomes important: since the evolution of u(T) and v(T) is assumed to be slow, they are treated as constant for the purpose of the Fourier transformation. Since we are interested in steady states, we drop the higher-order derivatives and rearrange the resulting equation to

ddT(uv)=18ωd(4v(ω02−ωd2)+3α(v3+u2v)−4Fsin⁡θ4u(ωd2−ω02)−3α(u3+uv2)−4Fcos⁡θ).

Steady states can now be found by setting the l.h.s. to zero, i.e., assuming u(T) and v(T) constant and neglecting any transient behaviour. This results in a set of 2 nonlinear polynomial equations of order 3, for which the maximum number of solutions set by Bézout's theorem is 32=9. Depending on the parameters, the number of real solutions is known to be between 1 and 3.

Sidenote: perturbative approach ​

The steady states describe a response that may be recast as x0(t)=X0cos⁡(ωdt+ϕ), where X0=u2+v2 and ϕ=−atan(v/u). Frequency conversion from ωd to 3ωd can be found by setting x(t)≡x0(t)+δx(t) with |δx(t)|≪|x0(t)| and expanding Eq. \eqref{eq:duffing} to first-order in δx(t). The resulting equation

δx¨(t)+[ω02+3αX024]δx(t)=−αX034cos⁡(3ωdt+3ϕ),

describes a simple harmonic oscillator, which is exactly soluble. Correspondingly, a response of δx(t) at frequency 3ωd is observed. Since this response is obtained 'on top of' each steady state of the equations of motion, no previously-unknown solutions are generated in the process.

Two-frequency ansatz ​

An approach in the spirit of harmonic balance is to use both harmonics ωd and 3ωd on the same footing, i.e., to insert the ansatz

x(t)=u1(T)cos⁡(ωdt)+v1(T)sin⁡(ωdt)+u2(T)cos⁡(3ωdt)+v2(T)sin⁡(3ωdt),

with u1,u2,v1,v2 being the harmonic variables. As before we substitute the ansatz into Eq. \eqref{eq:duffing}, drop second derivatives with respect to T and Fourier-transform both sides. Now, the respective coefficients correspond to cos⁡(ωdt), sin⁡(ωdt), cos⁡(3ωdt) and sin⁡(3ωdt). Rearranging, we obtain

du1dT=12ωd[(ω02−ωd2)v1+3α4(v13+u12v1+u12v2−v12v2+2u22v1+2v22v1−2u1u2v1)+Fsin⁡θ],dv1dT=12ωd[(ωd2−ω02)u1−3α4(u13+u12u2+v12u1−v12u2+2u22u1+2v22u1+2u1v1v2)−Fcos⁡θ],du2dT=16ωd[(ω02−9ωd2)v2+α4(−v13+3v23+3u12v1+6u12v2+3u22v2+6v12v2)],dv2dT=16ωd[(9ωd2−ω02)u2−α4(u13+3u23+6u12u2−3v12u1+3v22u2+6v12u2)].

In contrast to the single-frequency ansatz, we now have 4 equations of order 3, allowing up to 34=81 solutions (the number of unique real ones is again generally far smaller). The larger number of solutions is explained by higher harmonics which cannot be captured perturbatively by the single-frequency ansatz. In particular, those where the 3ωd component is significant. Such solutions appear, e.g., for ωd≈ω0/3 where the generated 3ωd harmonic is close to the natural resonant frequency. See the examples for numerical results.

- +
HarmonicBalance.jl

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The method of harmonic balance ​

Frequency conversion in oscillating nonlinear systems ​

HarmonicBalance.jl focuses on harmonically-driven nonlinear systems, i.e., dynamical systems governed by equations of motion where all explicitly time-dependent terms are harmonic. Let us take a general nonlinear system of N second-order ODEs with real variables xi(t), i=1,2,⋯,N and time t as the independent variable,

x¨(t)+F(x(t),t)=0.

The vector x(t)=(x1(t),...,xN(t))T fully describes the state of the system. Physically, x(t) encompasses the amplitudes of either point-like or collective oscillators (e.g., mechanical resonators, voltage oscillations in RLC circuits, an oscillating electrical dipole moment, or standing modes of an optical cavity).

As the simplest example, let us first solve the harmonic oscillator in frequency space. The equation of motion is

x¨(t)+γx˙(t)+ω02x(t)=Fcos⁡(ωdt)

where γ is the damping coefficient and ω0 the natural frequency. Fourier-transforming both sides of this equation gives

(ω02−ω2+iωγ)x~(ω)=F2[δ(ω+ωd)+δ(ω−ωd)].

Evidently, x~(ω) is only nonvanishing for ω=±ωd. The system thus responds at the driving frequency only - the behaviour can be captured by a single harmonic. This illustrates the general point that linear systems are exactly solvable by transforming to Fourier space, where the equations are diagonal.

The situation becomes more complex if nonlinear terms are present, as these cause frequency conversion. Suppose we add a quadratic nonlinearity βx2(t) to the equations of motion; an attempt to Fourier-transform gives

FT[x2](ω)=∫x2(t)e−iωtdt=∫−∞+∞x~(ω′)x~(ω″)δ(ω″+ω′−ω)dω′dω″,

which couples all harmonics ω,ω′,ω″ such that ω+ω′+ω″=0. To lowest order, this means the induced motion at the drive frequency generates a higher harmonic, ωd→2ωd. To higher orders however, the frequency conversion propagates through the spectrum, coupling an infinite number of harmonics. The system is not solvable in Fourier space anymore!

Harmonic ansatz & harmonic equations ​

Even though we need an infinity of Fourier components to describe our system exactly, some components are more important than others. The strategy of harmonic balance is to describe the motion of any variable xi(t) in a truncated Fourier space

xi(t)=∑j=1Miui,j(T)cos⁡(ωi,jt)+vi,j(T)sin⁡(ωi,jt).

Within this space, the system is described by a finite-dimensional vector

u(T)=(u1,1(T),v1,1(T),…uN,MN(T),vN,MN(T))

Under the assumption that u(T) evolves at much slower timescales than the oscillatory terms ωi,jt, we may neglect all of its higher order time derivatives. Notice that once ansatz \eqref{eq:harmansatz} is used in Eq. \eqref{eq:ode}, all terms become oscillatory - each prefactor of cos⁡(ωi,jt) and sin⁡(ωi,jt) thus generates a separate equation. Collecting these, we obtain a 1st order nonlinear ODEs,

du(T)dT=F¯(u),

which we call the harmonic equations. The main purpose of HarmonicBalance.jl is to obtain and solve them. We are primarily interested in steady states u0 defined by F¯(u0)=0.

The process of obtaining the harmonic equations is best shown on an example.

Example: the Duffing oscillator ​

Here, we derive the harmonic equations for a single Duffing resonator, governed by the equation

x¨(t)+ω02x(t)+αx3(t)=Fcos⁡(ωdt+θ).

As explained in above, for a periodic driving at frequency ωd and a weak nonlinearity α, we expect the response at frequency ωd to dominate, followed by a response at 3ωd due to frequency conversion.

Single-frequency ansatz ​

We first attempt to describe the steady states of Eq. \eqref{eq:duffing} using only one harmonic, ωd. The starting point is the harmonic ansatz for x

x(t)=u(T)cos⁡(ωdt)+v(T)sin⁡(ωdt),

with the harmonic variables u and v. The slow time T is, for now, equivalent to t. Substituting this ansatz into mechanical equations of motion results in

[u¨+2ωdv˙+u(ω02−ωd2)+3α(u3+uv2)4+Fcos⁡θ]cos⁡(ωdt)+[v¨−2ωdu˙+v(ω02−ωd2)+3α(v3+u2v)4−Fsin⁡θ]sin⁡(ωdt)+α(u3−3uv2)4cos⁡(3ωdt)+α(3u2v−v3)4sin⁡(3ωdt)=0.

We see that the x3 term has generated terms that oscillate at 3ωd, describing the process of frequency upconversion. We now Fourier-transform both sides of Eq. \eqref{eq:ansatz1} with respect to ωd to obtain the harmonic equations. This process is equivalent to extracting the respective coefficients of cos⁡(ωdt) and sin⁡(ωdt). Here the distinction between t and T becomes important: since the evolution of u(T) and v(T) is assumed to be slow, they are treated as constant for the purpose of the Fourier transformation. Since we are interested in steady states, we drop the higher-order derivatives and rearrange the resulting equation to

ddT(uv)=18ωd(4v(ω02−ωd2)+3α(v3+u2v)−4Fsin⁡θ4u(ωd2−ω02)−3α(u3+uv2)−4Fcos⁡θ).

Steady states can now be found by setting the l.h.s. to zero, i.e., assuming u(T) and v(T) constant and neglecting any transient behaviour. This results in a set of 2 nonlinear polynomial equations of order 3, for which the maximum number of solutions set by Bézout's theorem is 32=9. Depending on the parameters, the number of real solutions is known to be between 1 and 3.

Sidenote: perturbative approach ​

The steady states describe a response that may be recast as x0(t)=X0cos⁡(ωdt+ϕ), where X0=u2+v2 and ϕ=−atan(v/u). Frequency conversion from ωd to 3ωd can be found by setting x(t)≡x0(t)+δx(t) with |δx(t)|≪|x0(t)| and expanding Eq. \eqref{eq:duffing} to first-order in δx(t). The resulting equation

δx¨(t)+[ω02+3αX024]δx(t)=−αX034cos⁡(3ωdt+3ϕ),

describes a simple harmonic oscillator, which is exactly soluble. Correspondingly, a response of δx(t) at frequency 3ωd is observed. Since this response is obtained 'on top of' each steady state of the equations of motion, no previously-unknown solutions are generated in the process.

Two-frequency ansatz ​

An approach in the spirit of harmonic balance is to use both harmonics ωd and 3ωd on the same footing, i.e., to insert the ansatz

x(t)=u1(T)cos⁡(ωdt)+v1(T)sin⁡(ωdt)+u2(T)cos⁡(3ωdt)+v2(T)sin⁡(3ωdt),

with u1,u2,v1,v2 being the harmonic variables. As before we substitute the ansatz into Eq. \eqref{eq:duffing}, drop second derivatives with respect to T and Fourier-transform both sides. Now, the respective coefficients correspond to cos⁡(ωdt), sin⁡(ωdt), cos⁡(3ωdt) and sin⁡(3ωdt). Rearranging, we obtain

du1dT=12ωd[(ω02−ωd2)v1+3α4(v13+u12v1+u12v2−v12v2+2u22v1+2v22v1−2u1u2v1)+Fsin⁡θ],dv1dT=12ωd[(ωd2−ω02)u1−3α4(u13+u12u2+v12u1−v12u2+2u22u1+2v22u1+2u1v1v2)−Fcos⁡θ],du2dT=16ωd[(ω02−9ωd2)v2+α4(−v13+3v23+3u12v1+6u12v2+3u22v2+6v12v2)],dv2dT=16ωd[(9ωd2−ω02)u2−α4(u13+3u23+6u12u2−3v12u1+3v22u2+6v12u2)].

In contrast to the single-frequency ansatz, we now have 4 equations of order 3, allowing up to 34=81 solutions (the number of unique real ones is again generally far smaller). The larger number of solutions is explained by higher harmonics which cannot be captured perturbatively by the single-frequency ansatz. In particular, those where the 3ωd component is significant. Such solutions appear, e.g., for ωd≈ω0/3 where the generated 3ωd harmonic is close to the natural resonant frequency. See the examples for numerical results.

+ \ No newline at end of file diff --git a/dev/background/limit_cycles.html b/dev/background/limit_cycles.html index efa915c3..44101962 100644 --- a/dev/background/limit_cycles.html +++ b/dev/background/limit_cycles.html @@ -5,12 +5,13 @@ Limit cycles | HarmonicBalance.jl - - + + + - + - + @@ -21,8 +22,8 @@ -
HarmonicBalance.jl

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Limit cycles ​

We explain how HarmonicBalance.jl uses a new technique to find limit cycles in systems of nonlinear ODEs. For a more in depth overwiew see Chapter 6 in Jan Košata's PhD theses or del_Pino_2024.

Limit cycles from a Hopf bifurcation ​

The end product of the harmonic balance technique are what we call the harmonic equations, i.e., first-order ODEs for the harmonic variables U(T):

dU(T)dT=G―(U)

These Odes have no explicit time-dependence - they are autonomous. We have mostly been searching for steady states, which likewise show no time dependence. However, time-dependent solutions to autonomous ODEs can also exist. One mechanism for their creation is a Hopf bifurcation - a critical point where a stable solution transitions into an unstable one. For a stable solution, the associated eigenvalues λ of the linearisation all satisfy Re(λ)<0. When a Hopf bifurcation takes place, one complex-conjugate pair of eigenvalues crosses the real axis such that Re(λ)>0. The state is then, strictly speaking, unstable. However, instead of evolving into another steady state, the system may assume a periodic orbit in phase space, giving a solution of the form

U(T)=U0+Ulccos⁡(ωlcT+ϕ)

which is an example of a limit cycle. We denote the originating steady state as Hopf-unstable.

We can continue to use harmonic balance as the solution still describes a harmonic response Allwright (1977). If we translate back to the the lab frame [variable x(t)], clearly, each frequency ωj constituting our harmonic ansatz [U(T)], we obtain frequencies ωj as well as ωj±ωlc  in the lab frame. Furthermore, as multiple harmonics now co-exist in the system, frequency conversion may take place, spawning further pairs ωj±kωlc  with integer k. Therefore, to construct a harmonic ansatz capturing limit cycles, we simply add an integer number K of such pairs to our existing set of M harmonics,

{ω1,…,ωM}→{ω1,ω1±ωlc,ω1±2ωlc,…,ωM±Kωlc}

Ansatz ​

Original ansatz ​

Having seen how limit cycles are formed, we now proceed to tackle a key problem: how to find their frequency ωlc. We again demonstrate by considering a single variable x(t). We may try the simplest ansatz for a system driven at frequency ω,

x(t)=u1(T)cos⁡(ωt)+v1(T)sin⁡(ωt)

In this formulation, limit cycles may be obtained by solving the resulting harmonic equations with a Runge-Kutta type solver to obtain the time evolution of u1(T) and v1(T). See the limit cycle tutorial for an example.

Extended ansatz ​

Including newly-emergent pairs of harmonics is in principle straightforward. Suppose a limit cycle has formed in our system with a frequency ωlc, prompting the ansatz

x(t)=u1cos⁡(ωt)+v1sin⁡(ωt)+u2cos⁡[(ω+ωlc)t]+v2sin⁡[(ω+ωlc)t]+u3cos⁡[(ω−ωlc)t]+v3sin⁡[(ω−ωlc)t]+…

where each of the ω±kωlc  pairs contributes 4 harmonic variables. The limit cycle frequency ωlc is also a variable in this formulation, but does not contribute a harmonic equation, since dωlc/dT=0 by construction. We thus arrive at a total of 2+4K harmonic equations in 2+4K+1 variables. To obtain steady states, we must thus solve an underdetermined system, which has an infinite number of solutions. Given that we expect the limit cycles to possess U(1) gauge freedom, this is a sensible observation. We may still use iterative numerical procedures such as the Newton method to find solutions one by one, but homotopy continuation is not applicable. In this formulation, steady staes states are characterised by zero entries for u2,v2,…u2K+1,v2K+1. The variable ωlc  is redundant and may take any value - the states therefore also appear infinitely degenerate, which, however, has no physical grounds. Oppositely, solutions may appear for which some of the limit cycle variables u2,v2,…u2K+1,v2K+1 are nonzero, but ωlc =0. These violate our assumption of distinct harmonic variables corresponding to distinct frequencies and are therefore discarded.

Gauge fixing ​

We now constrain the system to remove the U(1) gauge freedom. This is best done by explicitly writing out the free phase. Recall that our solution must be symmetric under a time translation symmetry, that is, taking t→t+2π/ω. Applying this n times transforms x(t) into

x(t)=u1cos⁡(ωt)+v1sin⁡(ωt)+u2cos⁡[(ω+ωlc)t+ϕ]+v2sin⁡[(ω+ωlc)t+ϕ]+u3cos⁡[(ω−ωlc)t−ϕ]+v3sin⁡[(ω−ωlc)t−ϕ]+…

where we defined ϕ=2πnωlc /ω. Since ϕ is free, we can fix it to, for example,

ϕ=−arctan⁡u2/v2

which turns into

x(t)=u1cos⁡(ωt)+v1sin⁡(ωt)+(v2cos⁡ϕ−u2sin⁡ϕ)sin⁡[(ω+ωlc)t]+(u3cos⁡ϕ−v3sin⁡ϕ)cos⁡[(ω−ωlc)t]+(v3cos⁡ϕ+u3sin⁡ϕ)[(ω−ωlc)t]+…

We see that fixing the free phase has effectively removed one of the variables, since cos⁡[(ω+ωlc )t] does not appear any more. Discarding u2, we can therefore use 2+4K variables as our harmonic ansatz, i.e.,

U=(u1v1v2⋮v2K+1ωlc)

to remove the infinite degeneracy. Note that ϕ is only defined modulo π, but its effect on the harmonic variables is not. Choosing ϕ=−arctan⁡u2/v2+π would invert the signs of v2,u3,v3. As a result, each solution is doubly degenerate. Combined with the sign ambiguity of ωlc , we conclude that under the new ansatz, a limit cycle solution appears as a fourfold-degenerate steady state.

The harmonic equations can now be solved using homotopy continuation to obtain all steady states. Compared to the single-harmonic ansatz however, we have significantly enlarged the polynomial system to be solved. As the number of solutions scales exponentially (Bézout bound), we expect vast numbers of solutions even for fairly small systems.

- +
HarmonicBalance.jl

Appearance

Limit cycles ​

We explain how HarmonicBalance.jl uses a new technique to find limit cycles in systems of nonlinear ODEs. For a more in depth overwiew see Chapter 6 in Jan Košata's PhD theses or del_Pino_2024.

Limit cycles from a Hopf bifurcation ​

The end product of the harmonic balance technique are what we call the harmonic equations, i.e., first-order ODEs for the harmonic variables U(T):

dU(T)dT=G―(U)

These Odes have no explicit time-dependence - they are autonomous. We have mostly been searching for steady states, which likewise show no time dependence. However, time-dependent solutions to autonomous ODEs can also exist. One mechanism for their creation is a Hopf bifurcation - a critical point where a stable solution transitions into an unstable one. For a stable solution, the associated eigenvalues λ of the linearisation all satisfy Re(λ)<0. When a Hopf bifurcation takes place, one complex-conjugate pair of eigenvalues crosses the real axis such that Re(λ)>0. The state is then, strictly speaking, unstable. However, instead of evolving into another steady state, the system may assume a periodic orbit in phase space, giving a solution of the form

U(T)=U0+Ulccos⁡(ωlcT+ϕ)

which is an example of a limit cycle. We denote the originating steady state as Hopf-unstable.

We can continue to use harmonic balance as the solution still describes a harmonic response Allwright (1977). If we translate back to the the lab frame [variable x(t)], clearly, each frequency ωj constituting our harmonic ansatz [U(T)], we obtain frequencies ωj as well as ωj±ωlc  in the lab frame. Furthermore, as multiple harmonics now co-exist in the system, frequency conversion may take place, spawning further pairs ωj±kωlc  with integer k. Therefore, to construct a harmonic ansatz capturing limit cycles, we simply add an integer number K of such pairs to our existing set of M harmonics,

{ω1,…,ωM}→{ω1,ω1±ωlc,ω1±2ωlc,…,ωM±Kωlc}

Ansatz ​

Original ansatz ​

Having seen how limit cycles are formed, we now proceed to tackle a key problem: how to find their frequency ωlc. We again demonstrate by considering a single variable x(t). We may try the simplest ansatz for a system driven at frequency ω,

x(t)=u1(T)cos⁡(ωt)+v1(T)sin⁡(ωt)

In this formulation, limit cycles may be obtained by solving the resulting harmonic equations with a Runge-Kutta type solver to obtain the time evolution of u1(T) and v1(T). See the limit cycle tutorial for an example.

Extended ansatz ​

Including newly-emergent pairs of harmonics is in principle straightforward. Suppose a limit cycle has formed in our system with a frequency ωlc, prompting the ansatz

x(t)=u1cos⁡(ωt)+v1sin⁡(ωt)+u2cos⁡[(ω+ωlc)t]+v2sin⁡[(ω+ωlc)t]+u3cos⁡[(ω−ωlc)t]+v3sin⁡[(ω−ωlc)t]+…

where each of the ω±kωlc  pairs contributes 4 harmonic variables. The limit cycle frequency ωlc is also a variable in this formulation, but does not contribute a harmonic equation, since dωlc/dT=0 by construction. We thus arrive at a total of 2+4K harmonic equations in 2+4K+1 variables. To obtain steady states, we must thus solve an underdetermined system, which has an infinite number of solutions. Given that we expect the limit cycles to possess U(1) gauge freedom, this is a sensible observation. We may still use iterative numerical procedures such as the Newton method to find solutions one by one, but homotopy continuation is not applicable. In this formulation, steady staes states are characterised by zero entries for u2,v2,…u2K+1,v2K+1. The variable ωlc  is redundant and may take any value - the states therefore also appear infinitely degenerate, which, however, has no physical grounds. Oppositely, solutions may appear for which some of the limit cycle variables u2,v2,…u2K+1,v2K+1 are nonzero, but ωlc =0. These violate our assumption of distinct harmonic variables corresponding to distinct frequencies and are therefore discarded.

Gauge fixing ​

We now constrain the system to remove the U(1) gauge freedom. This is best done by explicitly writing out the free phase. Recall that our solution must be symmetric under a time translation symmetry, that is, taking t→t+2π/ω. Applying this n times transforms x(t) into

x(t)=u1cos⁡(ωt)+v1sin⁡(ωt)+u2cos⁡[(ω+ωlc)t+ϕ]+v2sin⁡[(ω+ωlc)t+ϕ]+u3cos⁡[(ω−ωlc)t−ϕ]+v3sin⁡[(ω−ωlc)t−ϕ]+…

where we defined ϕ=2πnωlc /ω. Since ϕ is free, we can fix it to, for example,

ϕ=−arctan⁡u2/v2

which turns into

x(t)=u1cos⁡(ωt)+v1sin⁡(ωt)+(v2cos⁡ϕ−u2sin⁡ϕ)sin⁡[(ω+ωlc)t]+(u3cos⁡ϕ−v3sin⁡ϕ)cos⁡[(ω−ωlc)t]+(v3cos⁡ϕ+u3sin⁡ϕ)[(ω−ωlc)t]+…

We see that fixing the free phase has effectively removed one of the variables, since cos⁡[(ω+ωlc )t] does not appear any more. Discarding u2, we can therefore use 2+4K variables as our harmonic ansatz, i.e.,

U=(u1v1v2⋮v2K+1ωlc)

to remove the infinite degeneracy. Note that ϕ is only defined modulo π, but its effect on the harmonic variables is not. Choosing ϕ=−arctan⁡u2/v2+π would invert the signs of v2,u3,v3. As a result, each solution is doubly degenerate. Combined with the sign ambiguity of ωlc , we conclude that under the new ansatz, a limit cycle solution appears as a fourfold-degenerate steady state.

The harmonic equations can now be solved using homotopy continuation to obtain all steady states. Compared to the single-harmonic ansatz however, we have significantly enlarged the polynomial system to be solved. As the number of solutions scales exponentially (Bézout bound), we expect vast numbers of solutions even for fairly small systems.

+ \ No newline at end of file diff --git a/dev/background/stability_response.html b/dev/background/stability_response.html index a843c751..b68292e9 100644 --- a/dev/background/stability_response.html +++ b/dev/background/stability_response.html @@ -5,12 +5,13 @@ Stability and linear response | HarmonicBalance.jl - - + + + - + - + @@ -21,8 +22,8 @@ -
HarmonicBalance.jl

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Stability and linear response ​

The core of the harmonic balance method is expressing the system's behaviour in terms of Fourier components or harmonics. For an N-coordinate system, we choose a set of Mi harmonics to describe each coordinate xi :

xi(t)=∑j=1Miui,j(T)cos⁡(ωi,jt)+vi,j(T)sin⁡(ωi,jt),

This means the system is now described using a discrete set of variables ui,j and vi,j. Constructing the vector

u(T)=(u1,1(T),v1,1(T),…uN,MN(T),vN,MN(T)),

we may obtain the harmonic equations (see an example of this procedure)

du(T)dT=F¯(u)

where F¯(u) is a nonlinear function. A steady state u0 is defined by F¯(u0)=0.

Stability ​

Let us assume that we found a steady state u0. When the system is in this state, it responds to small perturbations either by returning to u0 over some characteristic timescale (stable state) or by evolving away from u0 (unstable state). To analyze the stability of u0, we linearize the equations of motion around u0 for a small perturbation δu=u−u0 to obtain

ddT[δu(T)]=J(u0)δu(T),

where J(u0)=∇uF¯|u=u0 is the Jacobian matrix of the system evaluated at u=u0.

The linearised system is exactly solvable for δu(T) given an initial condition δu(T0). The solution can be expanded in terms of the complex eigenvalues λr and eigenvectors vr of J(u0), namely

δu(T)=∑rcrvreλrT.

The dynamical behaviour near the steady states is thus governed by eλrT: if Re(λr)<0 for all λr, the state u0 is stable. Conversely, if Re(λr)>0 for at least one λr, the state is unstable - perturbations such as noise or a small applied drive will force the system away from u0.

Linear response ​

The response of a stable steady state to an additional oscillatory force, caused by weak probes or noise, is often of interest. It can be calculated by solving for the perturbation δu(T) in the presence of an additional drive term.

ddT[δu(T)]=J(u0)δu(T)+ξeiΩT,

Suppose we have found an eigenvector of J(u0) such that J(u)v=λv. To solve the linearised equations of motion, we insert δu(T)=A(Ω)veiΩT. Projecting each side onto v gives

A(Ω)(iΩ−λ)=ξ⋅v⟹A(Ω)=ξ⋅v−Re[λ]+i(Ω−Im[λ])

We see that each eigenvalue λ results in a linear response that is a Lorentzian centered at Ω=Im[λ]. Effectively, the linear response matches that of a harmonic oscillator with resonance frequency Im[λ] and damping Re[λ].

Knowing the response of the harmonic variables u(T), what is the corresponding behaviour of the "natural" variables xi(t)? To find this out, we insert the perturbation back into the harmonic ansatz. Since we require real variables, let us use δu(T)=A(Ω)(veiΩT+v∗e−iΩT). Plugging this into

δxi(t)=∑j=1Miδui,j(t)cos⁡(ωi,jt)+δvi,j(t)sin⁡(ωi,jt)

and multiplying out the sines and cosines gives

δxi(t)=∑j=1Mi{(Re[δui,j]−Im[δvi,j])cos⁡[(ωi,j−Ω)t]+(Im[δui,j]+Re[δvi,j])sin⁡[(ωi,j−Ω)t]+(Re[δui,j]+Im[δvi,j])cos⁡[(ωi,j+Ω)t]+(−Im[δui,j]+Re[δvi,j])sin⁡[(ωi,j+Ω)t]}

where δui,j and δvi,j are the components of δu corresponding to the respective harmonics ωi,j.

We see that a motion of the harmonic variables at frequency Ω appears as motion of δxi(t) at frequencies ωi,j±Ω.

To make sense of this, we normalize the vector δu and use normalised components δu^i,j and δv^i,j. We also define the Lorentzian distribution

L(x)x0,γ=1(x−x0)2+γ2

We see that all components of δxi(t) are proportional to L(Ω)Im[λ],Re[λ]. The first and last two summands are Lorentzians centered at ±Ω which oscillate at ωi,j±Ω, respectively. From this, we can extract the linear response function in Fourier space, χ(ω~)

|χ[δxi](ω~)|2=∑j=1Mi{[(Re[δu^i,j]−Im[δv^i,j])2+(Im[δu^i,j]+Re[δv^i,j])2]L(ωi,j−ω~)Im[λ],Re[λ]+[(Re[δu^i,j]+Im[δv^i,j])2+(Re[δv^i,j]−Im[δu^i,j])2]L(ω~−ωi,j)Im[λ],Re[λ]}

Keeping in mind that L(x)x0,γ=L(x+Δ)x0+Δ,γ and the normalization δu^i,j2+δv^i,j2=1, we can rewrite this as

|χ[δxi](ω~)|2=∑j=1Mi(1+αi,j)L(ω~)ωi,j−Im[λ],Re[λ]+(1−αi,j)L(ω~)ωi,j+Im[λ],Re[λ]

where

αi,j=2(Im[δu^i,j]Re[δv^i,j]−Re[δu^i,j]Im[δv^i,j])

The above solution applies to every eigenvalue λ of the Jacobian. It is now clear that the linear response function χ[δxi](ω~) contains for each eigenvalue λr and harmonic ωi,j :

  • A Lorentzian centered at ωi,j−Im[λr] with amplitude 1+αi,j(r)

  • A Lorentzian centered at ωi,j+Im[λr] with amplitude 1−αi,j(r)

Sidenote: As J a real matrix, there is an eigenvalue λr∗ for each λr. The maximum number of peaks in the linear response is thus equal to the dimensionality of u(T).

The linear response of the system in the state u0 is thus fully specified by the complex eigenvalues and eigenvectors of J(u0). In HarmonicBalance.jl, the module LinearResponse creates a set of plottable Lorentzian objects to represent this.

Check out this example of the linear response module of HarmonicBalance.jl

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HarmonicBalance.jl

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Stability and linear response ​

The core of the harmonic balance method is expressing the system's behaviour in terms of Fourier components or harmonics. For an N-coordinate system, we choose a set of Mi harmonics to describe each coordinate xi :

xi(t)=∑j=1Miui,j(T)cos⁡(ωi,jt)+vi,j(T)sin⁡(ωi,jt),

This means the system is now described using a discrete set of variables ui,j and vi,j. Constructing the vector

u(T)=(u1,1(T),v1,1(T),…uN,MN(T),vN,MN(T)),

we may obtain the harmonic equations (see an example of this procedure)

du(T)dT=F¯(u)

where F¯(u) is a nonlinear function. A steady state u0 is defined by F¯(u0)=0.

Stability ​

Let us assume that we found a steady state u0. When the system is in this state, it responds to small perturbations either by returning to u0 over some characteristic timescale (stable state) or by evolving away from u0 (unstable state). To analyze the stability of u0, we linearize the equations of motion around u0 for a small perturbation δu=u−u0 to obtain

ddT[δu(T)]=J(u0)δu(T),

where J(u0)=∇uF¯|u=u0 is the Jacobian matrix of the system evaluated at u=u0.

The linearised system is exactly solvable for δu(T) given an initial condition δu(T0). The solution can be expanded in terms of the complex eigenvalues λr and eigenvectors vr of J(u0), namely

δu(T)=∑rcrvreλrT.

The dynamical behaviour near the steady states is thus governed by eλrT: if Re(λr)<0 for all λr, the state u0 is stable. Conversely, if Re(λr)>0 for at least one λr, the state is unstable - perturbations such as noise or a small applied drive will force the system away from u0.

Linear response ​

The response of a stable steady state to an additional oscillatory force, caused by weak probes or noise, is often of interest. It can be calculated by solving for the perturbation δu(T) in the presence of an additional drive term.

ddT[δu(T)]=J(u0)δu(T)+ξeiΩT,

Suppose we have found an eigenvector of J(u0) such that J(u)v=λv. To solve the linearised equations of motion, we insert δu(T)=A(Ω)veiΩT. Projecting each side onto v gives

A(Ω)(iΩ−λ)=ξ⋅v⟹A(Ω)=ξ⋅v−Re[λ]+i(Ω−Im[λ])

We see that each eigenvalue λ results in a linear response that is a Lorentzian centered at Ω=Im[λ]. Effectively, the linear response matches that of a harmonic oscillator with resonance frequency Im[λ] and damping Re[λ].

Knowing the response of the harmonic variables u(T), what is the corresponding behaviour of the "natural" variables xi(t)? To find this out, we insert the perturbation back into the harmonic ansatz. Since we require real variables, let us use δu(T)=A(Ω)(veiΩT+v∗e−iΩT). Plugging this into

δxi(t)=∑j=1Miδui,j(t)cos⁡(ωi,jt)+δvi,j(t)sin⁡(ωi,jt)

and multiplying out the sines and cosines gives

δxi(t)=∑j=1Mi{(Re[δui,j]−Im[δvi,j])cos⁡[(ωi,j−Ω)t]+(Im[δui,j]+Re[δvi,j])sin⁡[(ωi,j−Ω)t]+(Re[δui,j]+Im[δvi,j])cos⁡[(ωi,j+Ω)t]+(−Im[δui,j]+Re[δvi,j])sin⁡[(ωi,j+Ω)t]}

where δui,j and δvi,j are the components of δu corresponding to the respective harmonics ωi,j.

We see that a motion of the harmonic variables at frequency Ω appears as motion of δxi(t) at frequencies ωi,j±Ω.

To make sense of this, we normalize the vector δu and use normalised components δu^i,j and δv^i,j. We also define the Lorentzian distribution

L(x)x0,γ=1(x−x0)2+γ2

We see that all components of δxi(t) are proportional to L(Ω)Im[λ],Re[λ]. The first and last two summands are Lorentzians centered at ±Ω which oscillate at ωi,j±Ω, respectively. From this, we can extract the linear response function in Fourier space, χ(ω~)

|χ[δxi](ω~)|2=∑j=1Mi{[(Re[δu^i,j]−Im[δv^i,j])2+(Im[δu^i,j]+Re[δv^i,j])2]L(ωi,j−ω~)Im[λ],Re[λ]+[(Re[δu^i,j]+Im[δv^i,j])2+(Re[δv^i,j]−Im[δu^i,j])2]L(ω~−ωi,j)Im[λ],Re[λ]}

Keeping in mind that L(x)x0,γ=L(x+Δ)x0+Δ,γ and the normalization δu^i,j2+δv^i,j2=1, we can rewrite this as

|χ[δxi](ω~)|2=∑j=1Mi(1+αi,j)L(ω~)ωi,j−Im[λ],Re[λ]+(1−αi,j)L(ω~)ωi,j+Im[λ],Re[λ]

where

αi,j=2(Im[δu^i,j]Re[δv^i,j]−Re[δu^i,j]Im[δv^i,j])

The above solution applies to every eigenvalue λ of the Jacobian. It is now clear that the linear response function χ[δxi](ω~) contains for each eigenvalue λr and harmonic ωi,j :

  • A Lorentzian centered at ωi,j−Im[λr] with amplitude 1+αi,j(r)

  • A Lorentzian centered at ωi,j+Im[λr] with amplitude 1−αi,j(r)

Sidenote: As J a real matrix, there is an eigenvalue λr∗ for each λr. The maximum number of peaks in the linear response is thus equal to the dimensionality of u(T).

The linear response of the system in the state u0 is thus fully specified by the complex eigenvalues and eigenvectors of J(u0). In HarmonicBalance.jl, the module LinearResponse creates a set of plottable Lorentzian objects to represent this.

Check out this example of the linear response module of HarmonicBalance.jl

+ \ No newline at end of file diff --git a/dev/examples/index.html b/dev/examples/index.html index 8b96e0d3..6680c80e 100644 --- a/dev/examples/index.html +++ b/dev/examples/index.html @@ -5,12 +5,13 @@ Examples | HarmonicBalance.jl - - + + + - + - + @@ -22,8 +23,8 @@ - - + + \ No newline at end of file diff --git a/dev/examples/parametric_via_three_wave_mixing.html b/dev/examples/parametric_via_three_wave_mixing.html index 2cfe6574..2beed4a7 100644 --- a/dev/examples/parametric_via_three_wave_mixing.html +++ b/dev/examples/parametric_via_three_wave_mixing.html @@ -5,12 +5,13 @@ Parametric Pumping via Three-Wave Mixing | HarmonicBalance.jl - - + + + - + - + @@ -21,7 +22,7 @@ -
HarmonicBalance.jl

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Parametric Pumping via Three-Wave Mixing ​

julia
using HarmonicBalance, Plots
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HarmonicBalance.jl
Appearance
Parametric Pumping via Three-Wave Mixing ​
julia
using HarmonicBalance, Plots
 using Plots.Measures
 using Random
System ​
julia
@variables β α ω ω0 F γ t x(t) # declare constant variables and a function x(t)
 diff_eq = DifferentialEquation(
@@ -63,7 +64,7 @@
 method = TotalDegree()
 result = get_steady_states(harmonic_eq2, method, varied, fixed)
 plot_phase_diagram(result; class="stable")
This page was generated using Literate.jl.

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     Parametrically driven resonator | HarmonicBalance.jl
     
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HarmonicBalance.jl
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Parametrically driven resonator ​
One of the most famous effects displaced by nonlinear oscillators is parametric resonance, where the frequency of the linear resonator is modulated in time Phys. Rev. E 94, 022201 (2016). In the following we analyse this system, governed by the equations
x¨(t)+γx˙(t)+Ω2(1−λcos⁡(2ωt+ψ))x+αx3+ηx2x˙+Fd(t)=0
where for completeness we also considered an external drive term Fd(t)=Fcos⁡(ωt+θ) and a nonlinear damping term ηx2x˙
To implement this system in Harmonic Balance, we first import the library
julia
using HarmonicBalance
Subsequently, we type define parameters in the problem and the oscillating amplitude function x(t) using the variables macro from Symbolics.jl
julia
@variables ω₀ γ λ F η α ω t x(t)
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HarmonicBalance.jl
Appearance
Parametrically driven resonator ​
One of the most famous effects displaced by nonlinear oscillators is parametric resonance, where the frequency of the linear resonator is modulated in time Phys. Rev. E 94, 022201 (2016). In the following we analyse this system, governed by the equations
x¨(t)+γx˙(t)+Ω2(1−λcos⁡(2ωt+ψ))x+αx3+ηx2x˙+Fd(t)=0
where for completeness we also considered an external drive term Fd(t)=Fcos⁡(ωt+θ) and a nonlinear damping term ηx2x˙
To implement this system in Harmonic Balance, we first import the library
julia
using HarmonicBalance
Subsequently, we type define parameters in the problem and the oscillating amplitude function x(t) using the variables macro from Symbolics.jl
julia
@variables ω₀ γ λ F η α ω t x(t)
 
 natural_equation =
     d(d(x, t), t) +
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 plot!(result, "sqrt(u1^2 + v1^2)"; not_class="large")
Alternatively, we may visualise all underlying solutions, including complex ones,
julia
plot(result, "sqrt(u1^2 + v1^2)"; class="all")
2D parameters ​
The parametrically driven oscillator boasts a stability diagram called "Arnold's tongues" delineating zones where the oscillator is stable from those where it is exponentially unstable (if the nonlinearity was absence). We can retrieve this diagram by calculating the steady states as a function of external detuning δ=ωL−ω0 and the parametric drive strength λ.
To perform a 2D sweep over driving frequency ω and parametric drive strength λ, we keep fixed from before but include 2 variables in varied
julia
fixed = (ω₀ => 1.0, γ => 1e-2, F => 1e-3, α => 1.0, η => 0.3)
 varied = (ω => range(0.8, 1.2, 50), λ => range(0.001, 0.6, 50))
 result_2D = get_steady_states(harmonic_eq, varied, fixed);

-Solving for 2500 parameters...  50%|██████████▏         |  ETA: 0:00:01
-  # parameters solved:  1260
-  # paths tracked:      6300
+Solving for 2500 parameters...  52%|██████████▍         |  ETA: 0:00:01
+  # parameters solved:  1301
+  # paths tracked:      6505
 
 
 
 
 
 
-Solving for 2500 parameters...  79%|███████████████▊    |  ETA: 0:00:00
-  # parameters solved:  1971
-  # paths tracked:      9855
+Solving for 2500 parameters...  81%|████████████████▎   |  ETA: 0:00:00
+  # parameters solved:  2031
+  # paths tracked:      10155
 
 
 
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   # paths tracked:      12500
Now, we count the number of solutions for each point and represent the corresponding phase diagram in parameter space. This is done using plot_phase_diagram. Only counting stable solutions,
julia
plot_phase_diagram(result_2D; class="stable")
In addition to phase diagrams, we can plot functions of the solution. The syntax is identical to 1D plotting. Let us overlay 2 branches into a single plot,
julia
# overlay branches with different colors
 plot(result_2D, "sqrt(u1^2 + v1^2)"; branch=1, class="stable", camera=(60, -40))
 plot!(result_2D, "sqrt(u1^2 + v1^2)"; branch=2, class="stable", color=:red)
Note that solutions are ordered in parameter space according to their closest neighbors. Plots can again be limited to a given class (e.g stable solutions only) through the keyword argument class.
This page was generated using Literate.jl.

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     Three Wave Mixing vs four wave mixing | HarmonicBalance.jl
     
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HarmonicBalance.jl
Appearance
Three Wave Mixing vs four wave mixing ​
Packages ​
We load the following packages into our environment:
julia
using HarmonicBalance, Plots
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HarmonicBalance.jl
Appearance
Three Wave Mixing vs four wave mixing ​
Packages ​
We load the following packages into our environment:
julia
using HarmonicBalance, Plots
 using Plots.Measures
 using Random
 
@@ -75,7 +76,7 @@
 p2 = plot(result; y="√(u2^2+v2^2)", legend=:best, ylims=(-0.1, 0.1))
 p3 = plot(result; y="√(u3^2+v3^2)", legend=:best)
 plot(p1, p2, p3; layout=(1, 3), size=(900, 300), margin=5mm)
This page was generated using Literate.jl.

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-{"background_harmonic_balance.md":"C6F-4l91","background_limit_cycles.md":"D4Bgf8Oo","background_stability_response.md":"CRWuXdC5","examples_index.md":"CbaECHS4","examples_parametric_via_three_wave_mixing.md":"BQYubtvq","examples_parametron.md":"BBoEju4P","examples_wave_mixing.md":"C3jNkjp6","index.md":"FCE4d8xL","introduction_citation.md":"FFyK_Tsl","introduction_index.md":"B8BdJeyN","introduction_resources.md":"CM-Vaq6Q","manual_entering_eom.md":"DampzHDw","manual_extracting_harmonics.md":"CIOZjrwR","manual_krylov-bogoliubov_method.md":"OSi_uHs9","manual_linear_response.md":"CD1hVQFy","manual_methods.md":"qGkkhEfO","manual_plotting.md":"CMWmhlSv","manual_saving.md":"CiRX_Ceq","manual_solving_harmonics.md":"Cup8_oE9","manual_time_dependent.md":"BeG_NgwS","tutorials_classification.md":"CI0IAjP2","tutorials_index.md":"DVZkm59g","tutorials_limit_cycles.md":"DI6zYoBX","tutorials_linear_response.md":"BW8rfOcP","tutorials_steady_states.md":"B6UwsFn2","tutorials_time_dependent.md":"DgL_ZpIM"}
+{"background_harmonic_balance.md":"C6F-4l91","background_limit_cycles.md":"D4Bgf8Oo","background_stability_response.md":"CRWuXdC5","examples_index.md":"CbaECHS4","examples_parametric_via_three_wave_mixing.md":"BQYubtvq","examples_parametron.md":"ApYdxv5d","examples_wave_mixing.md":"C3jNkjp6","index.md":"FCE4d8xL","introduction_citation.md":"FFyK_Tsl","introduction_index.md":"B8BdJeyN","introduction_resources.md":"CM-Vaq6Q","manual_entering_eom.md":"Ybl7g0Mg","manual_extracting_harmonics.md":"hd0gVEbJ","manual_krylov-bogoliubov_method.md":"C4zEJfOC","manual_linear_response.md":"CafqEdgB","manual_methods.md":"DTOoMn0y","manual_plotting.md":"C0lzz6zg","manual_saving.md":"DG3AlMel","manual_solving_harmonics.md":"Bi1hM2V8","manual_time_dependent.md":"uGO7eGa5","tutorials_classification.md":"CI0IAjP2","tutorials_index.md":"DVZkm59g","tutorials_limit_cycles.md":"DI6zYoBX","tutorials_linear_response.md":"BW8rfOcP","tutorials_steady_states.md":"B6UwsFn2","tutorials_time_dependent.md":"DgL_ZpIM"}
diff --git a/dev/index.html b/dev/index.html
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     HarmonicBalance.jl
     
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HarmonicBalance.jl
Appearance
HarmonicBalance.jl
Efficient Floquet expansions for nonlinear driven systems
A Julia suite for nonlinear dynamics using harmonic balance
Getting Started
Tutorials
View on GitHub
HarmonicBalance.jl
Non-equilibrium steady states
Compute all stationary states in a one or two-dimensional parameter sweep.
Linear Response
Explore the fluctuations on top of the non-equilibrium steady states.
Limit Cycles
Find limit cycles involving many frequencies.

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HarmonicBalance.jl
Appearance
HarmonicBalance.jl
Efficient Floquet expansions for nonlinear driven systems
A Julia suite for nonlinear dynamics using harmonic balance
Getting Started
Tutorials
View on GitHub
HarmonicBalance.jl
Non-equilibrium steady states
Compute all stationary states in a one or two-dimensional parameter sweep.
Linear Response
Explore the fluctuations on top of the non-equilibrium steady states.
Limit Cycles
Find limit cycles involving many frequencies.

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     Citation | HarmonicBalance.jl
     
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HarmonicBalance.jl
Appearance
Citation ​
If you use HarmonicBalance.jl in your project, we kindly ask you to cite this paper, namely:
HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance, Jan Košata, Javier del Pino, Toni L. Heugel, Oded Zilberberg, SciPost Phys. Codebases 6 (2022)
The limit cycle finding algorithm is based on the work of this paper:
Limit cycles as stationary states of an extended harmonic balance ansatz J. del Pino, J. Košata, and O. Zilberberg, Phys. Rev. Res. 6, 033180 (2024).

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HarmonicBalance.jl
Appearance
Citation ​
If you use HarmonicBalance.jl in your project, we kindly ask you to cite this paper, namely:
HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance, Jan Košata, Javier del Pino, Toni L. Heugel, Oded Zilberberg, SciPost Phys. Codebases 6 (2022)
The limit cycle finding algorithm is based on the work of this paper:
Limit cycles as stationary states of an extended harmonic balance ansatz J. del Pino, J. Košata, and O. Zilberberg, Phys. Rev. Res. 6, 033180 (2024).

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     Installation | HarmonicBalance.jl
     
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HarmonicBalance.jl
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Installation ​
It is easy to install HarmonicBalance.jl as we are registered in the Julia General registry. You can simply run the following command in the Julia REPL:
julia
julia> using Pkg
+    
HarmonicBalance.jl
Appearance
Installation ​
It is easy to install HarmonicBalance.jl as we are registered in the Julia General registry. You can simply run the following command in the Julia REPL:
julia
julia> using Pkg
 julia> Pkg.add("HarmonicBalance")
or
julia
julia> ] # `]` should be pressed
 julia> Pkg.add("HarmonicBalance")
You can check which version you have installled with the command
julia
julia> ]
 julia> status HarmonicBalance
Getting Started ​
Let us find the steady states of an external driven Duffing oscillator with nonlinear damping. Its equation of motion is:
x¨(t)+γx˙(t)+ω02x(t)⏟damped harmonic oscillator+αx(t)3⏟Duffing coefficient=Fcos⁡(ωt)⏟periodic drive
julia
using HarmonicBalance
@@ -42,7 +43,7 @@
    of which stable:  2
 
 Classes: stable, physical, Hopf, binary_labels
The obtained steady states can be plotted as a function of the driving frequency:
julia
plot(result, "sqrt(u1^2 + v1^2)")
If you want learn more on what you can do with HarmonicBalance.jl, check out the tutorials. We also have collected some examples of different physical systems.

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     Krylov-Bogoliubov Averaging Method | HarmonicBalance.jl
     
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HarmonicBalance.jl
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Krylov-Bogoliubov Averaging Method ​
The Krylov-Bogoliubov averaging method is an alternative high-frequency expansion technique used to analyze dynamical systems. Unlike the Harmonic Balance method, which is detailed in the background section, the Krylov-Bogoliubov method excels in computing higher orders in 1/ω, enabling the capture of faster dynamics within a system.
Purpose and Advantages ​
The primary advantage of the Krylov-Bogoliubov method lies in its ability to delve deeper into high-frequency components, allowing a more comprehensive understanding of fast dynamical behaviors. By leveraging this technique, one can obtain higher-order approximations that shed light on intricate system dynamics.
However, it's essential to note a limitation: this method cannot handle multiple harmonics within a single variable, unlike some other high-frequency expansion methods.
Usage ​
To compute the Krylov-Bogoliubov averaging method within your system, utilize the function get_krylov_equations. This function is designed specifically to implement the methodology and derive the equations necessary to analyze the system dynamics using this technique.
Function Reference ​
HarmonicBalance.KrylovBogoliubov.get_krylov_equations Function
julia
get_krylov_equations(
+    
HarmonicBalance.jl
Appearance
Krylov-Bogoliubov Averaging Method ​
The Krylov-Bogoliubov averaging method is an alternative high-frequency expansion technique used to analyze dynamical systems. Unlike the Harmonic Balance method, which is detailed in the background section, the Krylov-Bogoliubov method excels in computing higher orders in 1/ω, enabling the capture of faster dynamics within a system.
Purpose and Advantages ​
The primary advantage of the Krylov-Bogoliubov method lies in its ability to delve deeper into high-frequency components, allowing a more comprehensive understanding of fast dynamical behaviors. By leveraging this technique, one can obtain higher-order approximations that shed light on intricate system dynamics.
However, it's essential to note a limitation: this method cannot handle multiple harmonics within a single variable, unlike some other high-frequency expansion methods.
Usage ​
To compute the Krylov-Bogoliubov averaging method within your system, utilize the function get_krylov_equations. This function is designed specifically to implement the methodology and derive the equations necessary to analyze the system dynamics using this technique.
Function Reference ​
HarmonicBalance.KrylovBogoliubov.get_krylov_equations Function
julia
get_krylov_equations(
     diff_eom::DifferentialEquation;
     order,
     fast_time,
@@ -49,8 +50,8 @@
 
 ((1//2)*(ω^2)*v1(T) - (1//2)*(ω0^2)*v1(T)) / ω ~ Differential(T)(u1(T))
 
-((1//2)*(ω0^2)*u1(T) - (1//2)*F - (1//2)*(ω^2)*u1(T)) / ω ~ Differential(T)(v1(T))
source
For further information and a detailed understanding of this method, refer to Krylov-Bogoliubov averaging method on Wikipedia.

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+((1//2)*(ω0^2)*u1(T) - (1//2)*F - (1//2)*(ω^2)*u1(T)) / ω ~ Differential(T)(v1(T))
source
For further information and a detailed understanding of this method, refer to Krylov-Bogoliubov averaging method on Wikipedia.

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diff --git a/dev/manual/entering_eom.html b/dev/manual/entering_eom.html
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     Entering equations of motion | HarmonicBalance.jl
     
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HarmonicBalance.jl
Appearance
Entering equations of motion ​
The struct DifferentialEquation is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using add_harmonic!.
DifferentialEquation.equations stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.
HarmonicBalance.DifferentialEquation Type
julia
mutable struct DifferentialEquation
Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for HarmonicBalance.jl ; after inputting the equations, the harmonics ansatz needs to be specified using add_harmonic!.
Fields
  • equations::OrderedCollections.OrderedDict{Num, Equation}: Assigns to each variable an equation of motion.
  • harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}: Assigns to each variable a set of harmonics.
Example
julia
julia> @variables t, x(t), y(t), ω0, ω, F, k;
+    
HarmonicBalance.jl
Appearance
Entering equations of motion ​
The struct DifferentialEquation is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using add_harmonic!.
DifferentialEquation.equations stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.
HarmonicBalance.DifferentialEquation Type
julia
mutable struct DifferentialEquation
Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for HarmonicBalance.jl ; after inputting the equations, the harmonics ansatz needs to be specified using add_harmonic!.
Fields
  • equations::OrderedCollections.OrderedDict{Num, Equation}: Assigns to each variable an equation of motion.
  • harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}: Assigns to each variable a set of harmonics.
Example
julia
julia> @variables t, x(t), y(t), ω0, ω, F, k;
 
 # equivalent ways to enter the simple harmonic oscillator
 julia> DifferentialEquation(d(x,t,2) + ω0^2 * x - F * cos(ω*t), x);
 julia> DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x);
 
 # two coupled oscillators, one of them driven
-julia> DifferentialEquation([d(x,t,2) + ω0^2 * x - k*y, d(y,t,2) + ω0^2 * y - k*x] .~ [F * cos(ω*t), 0], [x,y]);
source
HarmonicBalance.add_harmonic! Function
julia
add_harmonic!(diff_eom::DifferentialEquation, var::Num, ω)
Add the harmonic ω to the harmonic ansatz used to expand the variable var in diff_eom.
Example
define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω
julia
julia> @variables t, x(t), y(t), ω0, ω, F, k;
+julia> DifferentialEquation([d(x,t,2) + ω0^2 * x - k*y, d(y,t,2) + ω0^2 * y - k*x] .~ [F * cos(ω*t), 0], [x,y]);
source
HarmonicBalance.add_harmonic! Function
julia
add_harmonic!(diff_eom::DifferentialEquation, var::Num, ω)
Add the harmonic ω to the harmonic ansatz used to expand the variable var in diff_eom.
Example
define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω
julia
julia> @variables t, x(t), y(t), ω0, ω, F, k;
 julia> diff_eq = DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x);
 julia> add_harmonic!(diff_eq, x, ω) # expand x using ω
 
@@ -36,10 +37,10 @@
 Variables:       x(t)
 Harmonic ansatz: x(t) => ω;
 
-(ω0^2)*x(t) + Differential(t)(Differential(t)(x(t))) ~ F*cos(t*ω)
source
Symbolics.get_variables Method
julia
get_variables(diff_eom::DifferentialEquation) -> Vector{Num}
Return the dependent variables of diff_eom.
source
HarmonicBalance.get_independent_variables Method
julia
get_independent_variables(
+(ω0^2)*x(t) + Differential(t)(Differential(t)(x(t))) ~ F*cos(t*ω)
source
Symbolics.get_variables Method
julia
get_variables(diff_eom::DifferentialEquation) -> Vector{Num}
Return the dependent variables of diff_eom.
source
HarmonicBalance.get_independent_variables Method
julia
get_independent_variables(
     diff_eom::DifferentialEquation
-) -> Any
Return the independent dependent variables of diff_eom.
source

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Return the independent dependent variables of diff_eom.
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diff --git a/dev/manual/extracting_harmonics.html b/dev/manual/extracting_harmonics.html
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     Extracting harmonic equations | HarmonicBalance.jl
     
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HarmonicBalance.jl
Appearance
Extracting harmonic equations ​
Harmonic Balance method ​
Once a DifferentialEquation is defined and its harmonics specified, one can extract the harmonic equations using get_harmonic_equations, which itself is composed of the subroutines harmonic_ansatz, slow_flow, fourier_transform! and drop_powers.
The harmonic equations use an additional time variable specified as slow_time in get_harmonic_equations. This is essentially a label distinguishing the time dependence of the harmonic variables (expected to be slow) from that of the oscillating terms (expeted to be fast). When the equations are Fourier-transformed to remove oscillating terms, slow_time is treated as a constant. Such an approach is exact when looking for steady states.
HarmonicBalance.get_harmonic_equations Function
julia
get_harmonic_equations(diff_eom::DifferentialEquation; fast_time=nothing, slow_time=nothing)
Apply the harmonic ansatz, followed by the slow-flow, Fourier transform and dropping higher-order derivatives to obtain a set of ODEs (the harmonic equations) governing the harmonics of diff_eom.
The harmonics evolve in slow_time, the oscillating terms themselves in fast_time. If no input is used, a variable T is defined for slow_time and fast_time is taken as the independent variable of diff_eom.
By default, all products of order > 1 of slow_time-derivatives are dropped, which means the equations are linear in the time-derivatives.
Example
julia
julia> @variables t, x(t), ω0, ω, F;
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HarmonicBalance.jl
Appearance
Extracting harmonic equations ​
Harmonic Balance method ​
Once a DifferentialEquation is defined and its harmonics specified, one can extract the harmonic equations using get_harmonic_equations, which itself is composed of the subroutines harmonic_ansatz, slow_flow, fourier_transform! and drop_powers.
The harmonic equations use an additional time variable specified as slow_time in get_harmonic_equations. This is essentially a label distinguishing the time dependence of the harmonic variables (expected to be slow) from that of the oscillating terms (expeted to be fast). When the equations are Fourier-transformed to remove oscillating terms, slow_time is treated as a constant. Such an approach is exact when looking for steady states.
HarmonicBalance.get_harmonic_equations Function
julia
get_harmonic_equations(diff_eom::DifferentialEquation; fast_time=nothing, slow_time=nothing)
Apply the harmonic ansatz, followed by the slow-flow, Fourier transform and dropping higher-order derivatives to obtain a set of ODEs (the harmonic equations) governing the harmonics of diff_eom.
The harmonics evolve in slow_time, the oscillating terms themselves in fast_time. If no input is used, a variable T is defined for slow_time and fast_time is taken as the independent variable of diff_eom.
By default, all products of order > 1 of slow_time-derivatives are dropped, which means the equations are linear in the time-derivatives.
Example
julia
julia> @variables t, x(t), ω0, ω, F;
 
 # enter the simple harmonic oscillator
 julia> diff_eom = DifferentialEquation( d(x,t,2) + ω0^2 * x ~ F *cos(ω*t), x);
@@ -43,17 +44,17 @@
 
 (ω0^2)*u1(T) + (2//1)*ω*Differential(T)(v1(T)) - (ω^2)*u1(T) ~ F
 
-(ω0^2)*v1(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) ~ 0
source
HarmonicBalance.harmonic_ansatz Function
julia
harmonic_ansatz(eom::DifferentialEquation, time::Num; coordinates="Cartesian")
Expand each variable of diff_eom using the harmonics assigned to it with time as the time variable. For each harmonic of each variable, instance(s) of HarmonicVariable are automatically created and named.
source
HarmonicBalance.slow_flow Function
julia
slow_flow(eom::HarmonicEquation; fast_time::Num, slow_time::Num, degree=2)
Removes all derivatives w.r.t fast_time (and their products) in eom of power degree. In the remaining derivatives, fast_time is replaced by slow_time.
source
HarmonicBalance.fourier_transform Function
julia
fourier_transform(
+(ω0^2)*v1(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) ~ 0
source
HarmonicBalance.harmonic_ansatz Function
julia
harmonic_ansatz(eom::DifferentialEquation, time::Num; coordinates="Cartesian")
Expand each variable of diff_eom using the harmonics assigned to it with time as the time variable. For each harmonic of each variable, instance(s) of HarmonicVariable are automatically created and named.
source
HarmonicBalance.slow_flow Function
julia
slow_flow(eom::HarmonicEquation; fast_time::Num, slow_time::Num, degree=2)
Removes all derivatives w.r.t fast_time (and their products) in eom of power degree. In the remaining derivatives, fast_time is replaced by slow_time.
source
HarmonicBalance.fourier_transform Function
julia
fourier_transform(
     eom::HarmonicEquation,
     time::Num
-) -> HarmonicEquation
Extract the Fourier components of eom corresponding to the harmonics specified in eom.variables. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated time does not appear in the resulting equations anymore.
Underlying assumption: all time-dependences are harmonic.
source
HarmonicBalance.ExprUtils.drop_powers Function
julia
drop_powers(expr, vars, deg)
Remove parts of expr where the combined power of vars is => deg.
Example
julia
julia> @variables x,y;
+) -> HarmonicEquation
Extract the Fourier components of eom corresponding to the harmonics specified in eom.variables. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated time does not appear in the resulting equations anymore.
Underlying assumption: all time-dependences are harmonic.
source
HarmonicBalance.ExprUtils.drop_powers Function
julia
drop_powers(expr, vars, deg)
Remove parts of expr where the combined power of vars is => deg.
Example
julia
julia> @variables x,y;
 julia>drop_powers((x+y)^2, x, 2)
 y^2 + 2*x*y
 julia>drop_powers((x+y)^2, [x,y], 2)
 0
 julia>drop_powers((x+y)^2 + (x+y)^3, [x,y], 3)
-x^2 + y^2 + 2*x*y
source
HarmonicVariable and HarmonicEquation types ​
The equations governing the harmonics are stored using the two following structs. When going from the original to the harmonic equations, the harmonic ansatz xi(t)=∑j=1Mui,j(T)cos⁡(ωi,jt)+vi,j(T)sin⁡(ωi,jt) is used. Internally, each pair (ui,j,vi,j) is stored as a HarmonicVariable. This includes the identification of ωi,j and xi(t), which is needed to later reconstruct xi(t).
HarmonicBalance.HarmonicVariable Type
julia
mutable struct HarmonicVariable
Holds a variable stored under symbol describing the harmonic ω of natural_variable.
Fields
  • symbol::Num: Symbol of the variable in the HarmonicBalance namespace.
  • name::String: Human-readable labels of the variable, used for plotting.
  • type::String: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)
  • ω::Num: The harmonic being described.
  • natural_variable::Num: The natural variable whose harmonic is being described.
source
When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a HarmonicEquation. For an initial equation of motion consisting of M variables, each expanded in N harmonics, the resulting HarmonicEquation holds 2NM equations of 2NM variables. Each symbol not corresponding to a variable is identified as a parameter.
A HarmonicEquation can be either parsed into a steady-state Problem or solved using a dynamical ODE solver.
HarmonicBalance.HarmonicEquation Type
julia
mutable struct HarmonicEquation
Holds a set of algebraic equations governing the harmonics of a DifferentialEquation.
Fields
  • equations::Vector{Equation}: A set of equations governing the harmonics.
  • variables::Vector{HarmonicVariable}: A set of variables describing the harmonics.
  • parameters::Vector{Num}: The parameters of the equation set.
  • natural_equation::DifferentialEquation: The natural equation (before the harmonic ansatz was used).
source

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+x^2 + y^2 + 2*x*y
source
HarmonicVariable and HarmonicEquation types ​
The equations governing the harmonics are stored using the two following structs. When going from the original to the harmonic equations, the harmonic ansatz xi(t)=∑j=1Mui,j(T)cos⁡(ωi,jt)+vi,j(T)sin⁡(ωi,jt) is used. Internally, each pair (ui,j,vi,j) is stored as a HarmonicVariable. This includes the identification of ωi,j and xi(t), which is needed to later reconstruct xi(t).
HarmonicBalance.HarmonicVariable Type
julia
mutable struct HarmonicVariable
Holds a variable stored under symbol describing the harmonic ω of natural_variable.
Fields
  • symbol::Num: Symbol of the variable in the HarmonicBalance namespace.
  • name::String: Human-readable labels of the variable, used for plotting.
  • type::String: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)
  • ω::Num: The harmonic being described.
  • natural_variable::Num: The natural variable whose harmonic is being described.
source
When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a HarmonicEquation. For an initial equation of motion consisting of M variables, each expanded in N harmonics, the resulting HarmonicEquation holds 2NM equations of 2NM variables. Each symbol not corresponding to a variable is identified as a parameter.
A HarmonicEquation can be either parsed into a steady-state Problem or solved using a dynamical ODE solver.
HarmonicBalance.HarmonicEquation Type
julia
mutable struct HarmonicEquation
Holds a set of algebraic equations governing the harmonics of a DifferentialEquation.
Fields
  • equations::Vector{Equation}: A set of equations governing the harmonics.
  • variables::Vector{HarmonicVariable}: A set of variables describing the harmonics.
  • parameters::Vector{Num}: The parameters of the equation set.
  • natural_equation::DifferentialEquation: The natural equation (before the harmonic ansatz was used).
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diff --git a/dev/manual/linear_response.html b/dev/manual/linear_response.html
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     Linear response (WIP) | HarmonicBalance.jl
     
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HarmonicBalance.jl
Appearance
Linear response (WIP) ​
This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.
The methodology used is explained in Jan Kosata phd thesis.
Stability ​
The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,
HarmonicBalance.LinearResponse.get_Jacobian Function
julia
get_Jacobian(eom)
Obtain the symbolic Jacobian matrix of eom (either a HarmonicEquation or a DifferentialEquation). This is the linearised left-hand side of F(u) = du/dT.
source
Obtain a Jacobian from a DifferentialEquation by first converting it into a HarmonicEquation.
source
Get the Jacobian of a set of equations eqs with respect to the variables vars.
source
Linear response ​
The response to white noise can be shown with plot_linear_response. Depending on the order argument, different methods are used.
HarmonicBalance.LinearResponse.plot_linear_response Function
julia
plot_linear_response(res::Result, nat_var::Num; Ω_range, branch::Int, order=1, logscale=false, show_progress=true, kwargs...)
Plot the linear response to white noise of the variable nat_var for Result res on branch for input frequencies Ω_range. Slow-time derivatives up to order are kept in the process.
Any kwargs are fed to Plots' gr().
Solutions not belonging to the physical class are ignored.
source
First order ​
The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues λ describes a Lorentzian peak in the response; Re[λ] gives its center and Im[λ] its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment.
The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).
Behind the scenes, the spectra are stored using the dedicated structs Lorentzian and JacobianSpectrum.
HarmonicBalance.LinearResponse.JacobianSpectrum Type
julia
mutable struct JacobianSpectrum
Holds a set of Lorentzian objects belonging to a variable.
Fields
  • peaks::Vector{HarmonicBalance.LinearResponse.Lorentzian}
Constructor
julia
JacobianSpectrum(res::Result; index::Int, branch::Int)
source
HarmonicBalance.LinearResponse.Lorentzian Type
julia
struct Lorentzian
Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).
Fields
  • ω0::Float64
  • Γ::Float64
  • A::Float64
source
Higher orders ​
Setting order > 1 increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.
HarmonicBalance.LinearResponse.ResponseMatrix Type
julia
struct ResponseMatrix
Holds the compiled response matrix of a system.
Fields
  • matrix::Matrix{Function}: The response matrix (compiled).
  • symbols::Vector{Num}: Any symbolic variables in matrix to be substituted at evaluation.
  • variables::Vector{HarmonicVariable}: The frequencies of the harmonic variables underlying matrix. These are needed to transform the harmonic variables to the non-rotating frame.
source
HarmonicBalance.LinearResponse.get_response Function
julia
get_response(
+    
HarmonicBalance.jl
Appearance
Linear response (WIP) ​
This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.
The methodology used is explained in Jan Kosata phd thesis.
Stability ​
The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,
HarmonicBalance.LinearResponse.get_Jacobian Function
julia
get_Jacobian(eom)
Obtain the symbolic Jacobian matrix of eom (either a HarmonicEquation or a DifferentialEquation). This is the linearised left-hand side of F(u) = du/dT.
source
Obtain a Jacobian from a DifferentialEquation by first converting it into a HarmonicEquation.
source
Get the Jacobian of a set of equations eqs with respect to the variables vars.
source
Linear response ​
The response to white noise can be shown with plot_linear_response. Depending on the order argument, different methods are used.
HarmonicBalance.LinearResponse.plot_linear_response Function
julia
plot_linear_response(res::Result, nat_var::Num; Ω_range, branch::Int, order=1, logscale=false, show_progress=true, kwargs...)
Plot the linear response to white noise of the variable nat_var for Result res on branch for input frequencies Ω_range. Slow-time derivatives up to order are kept in the process.
Any kwargs are fed to Plots' gr().
Solutions not belonging to the physical class are ignored.
source
First order ​
The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues λ describes a Lorentzian peak in the response; Re[λ] gives its center and Im[λ] its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment.
The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).
Behind the scenes, the spectra are stored using the dedicated structs Lorentzian and JacobianSpectrum.
HarmonicBalance.LinearResponse.JacobianSpectrum Type
julia
mutable struct JacobianSpectrum
Holds a set of Lorentzian objects belonging to a variable.
Fields
  • peaks::Vector{HarmonicBalance.LinearResponse.Lorentzian}
Constructor
julia
JacobianSpectrum(res::Result; index::Int, branch::Int)
source
HarmonicBalance.LinearResponse.Lorentzian Type
julia
struct Lorentzian
Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).
Fields
  • ω0::Float64
  • Γ::Float64
  • A::Float64
source
Higher orders ​
Setting order > 1 increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.
HarmonicBalance.LinearResponse.ResponseMatrix Type
julia
struct ResponseMatrix
Holds the compiled response matrix of a system.
Fields
  • matrix::Matrix{Function}: The response matrix (compiled).
  • symbols::Vector{Num}: Any symbolic variables in matrix to be substituted at evaluation.
  • variables::Vector{HarmonicVariable}: The frequencies of the harmonic variables underlying matrix. These are needed to transform the harmonic variables to the non-rotating frame.
source
HarmonicBalance.LinearResponse.get_response Function
julia
get_response(
     rmat::HarmonicBalance.LinearResponse.ResponseMatrix,
     s::OrderedCollections.OrderedDict{Num, ComplexF64},
     Ω
-) -> Any
For rmat and a solution dictionary s, calculate the total response to a perturbative force at frequency Ω.
source
HarmonicBalance.LinearResponse.get_response_matrix Function
julia
get_response_matrix(diff_eq::DifferentialEquation, freq::Num; order=2)
Obtain the symbolic linear response matrix of a diff_eq corresponding to a perturbation frequency freq. This routine cannot accept a HarmonicEquation since there, some time-derivatives are already dropped. order denotes the highest differential order to be considered.
source

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+) -> Any
For rmat and a solution dictionary s, calculate the total response to a perturbative force at frequency Ω.
source
HarmonicBalance.LinearResponse.get_response_matrix Function
julia
get_response_matrix(diff_eq::DifferentialEquation, freq::Num; order=2)
Obtain the symbolic linear response matrix of a diff_eq corresponding to a perturbation frequency freq. This routine cannot accept a HarmonicEquation since there, some time-derivatives are already dropped. order denotes the highest differential order to be considered.
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HarmonicBalance.jl
Appearance
Methods ​
We offer several methods for solving the nonlinear algebraic equations that arise from the harmonic balance procedure. Each method has different tradeoffs between speed, robustness, and completeness.
Total Degree Method ​
HarmonicBalance.TotalDegree Type
julia
TotalDegree
The Total Degree homotopy method. Performs a homotopy H(x,t)=γtG(x)+(1−t)F(x) from the trivial polynomial system xᵢdᵢ+aᵢ with the maximal degree dᵢ determined by the Bezout bound. The method guarantees to find all solutions, however, it comes with a high computational cost. See HomotopyContinuation.jl for more information.
Fields
  • gamma::Complex: Complex multiplying factor of the start system G(x) for the homotopy
  • thread::Bool: Boolean indicating if threading is enabled.
  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.
  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.
  • compile::Union{Bool, Symbol}: Compilation options.
  • seed::UInt32: Seed for random number generation.
source
Polyhedral Method ​
HarmonicBalance.Polyhedral Type
julia
Polyhedral
The Polyhedral homotopy method. This method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (only_non_zero = true), resulting in speed up. See HomotopyContinuation.jl for more information.
Fields
  • only_non_zero::Bool: Boolean indicating if only non-zero solutions are considered.
  • thread::Bool: Boolean indicating if threading is enabled.
  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.
  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.
  • compile::Union{Bool, Symbol}: Compilation options.
  • seed::UInt32: Seed for random number generation.
source
Warm Up Method ​
HarmonicBalance.WarmUp Type
julia
WarmUp
The Warm Up method. This method prepares a warmup system using the parameter at index perturbed by perturbation_size and performs a homotopy using the warmup system to all other systems in the parameter sweep. It is very efficient for systems with less bifurcation in the parameter sweep. The Warm Up method does not guarantee to find all solutions. See HomotopyContinuation.jl for more information.
Fields
  • perturbation_size::ComplexF64: Size of the perturbation.
  • index::Union{Int64, EndpointRanges.Endpoint}: Index for the endpoint.
  • thread::Bool: Boolean indicating if threading is enabled.
  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.
  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.
  • compile::Union{Bool, Symbol}: Compilation options.
  • seed::UInt32: Seed for random number generation.
source

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HarmonicBalance.jl
Appearance
Methods ​
We offer several methods for solving the nonlinear algebraic equations that arise from the harmonic balance procedure. Each method has different tradeoffs between speed, robustness, and completeness.
Total Degree Method ​
HarmonicBalance.TotalDegree Type
julia
TotalDegree
The Total Degree homotopy method performs a homotopy H(x,t)=γtG(x)+(1−t)F(x) from the trivial polynomial system F(x)=xᵢdᵢ+aᵢ with the maximal degree dᵢ determined by the Bezout bound. The method guarantees to find all solutions, however, it comes with a high computational cost. See HomotopyContinuation.jl for more information.
Fields
  • gamma::Complex: Complex multiplying factor of the start system G(x) for the homotopy
  • thread::Bool: Boolean indicating if threading is enabled.
  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.
  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.
  • compile::Union{Bool, Symbol}: Compilation options.
  • seed::UInt32: Seed for random number generation.
source
Polyhedral Method ​
HarmonicBalance.Polyhedral Type
julia
Polyhedral
The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (only_non_zero = true), resulting in a speed up. See HomotopyContinuation.jl for more information.
Fields
  • only_non_zero::Bool: Boolean indicating if only non-zero solutions are considered.
  • thread::Bool: Boolean indicating if threading is enabled.
  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.
  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.
  • compile::Union{Bool, Symbol}: Compilation options.
  • seed::UInt32: Seed for random number generation.
source
Warm Up Method ​
HarmonicBalance.WarmUp Type
julia
WarmUp
The Warm Up method prepares a warmup system with the Total Degree method using the parameter at index perturbed by perturbation_size. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method does not guarantee to find all solutions. See HomotopyContinuation.jl for more information.
Fields
  • perturbation_size::ComplexF64: Size of the perturbation.
  • index::Union{Int64, EndpointRanges.Endpoint}: Index for the parameter set used as start system.
  • thread::Bool: Boolean indicating if threading is enabled.
  • tracker_options::HomotopyContinuation.TrackerOptions: Options for the tracker.
  • endgame_options::HomotopyContinuation.EndgameOptions: Options for the endgame.
  • compile::Union{Bool, Symbol}: Compilation options.
  • seed::UInt32: Seed for random number generation.
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diff --git a/dev/manual/plotting.html b/dev/manual/plotting.html
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     Analysis and plotting | HarmonicBalance.jl
     
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HarmonicBalance.jl
Appearance
Analysis and plotting ​
The key method for visualization is transform_solutions, which parses a string into a symbolic expression and evaluates it for every steady state solution.
HarmonicBalance.transform_solutions Function
julia
transform_solutions(
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HarmonicBalance.jl
Appearance
Analysis and plotting ​
The key method for visualization is transform_solutions, which parses a string into a symbolic expression and evaluates it for every steady state solution.
HarmonicBalance.transform_solutions Function
julia
transform_solutions(
     res::HarmonicBalance.Result,
     func;
     branches,
     realify
-) -> Vector
Takes a Result object and a string f representing a Symbolics.jl expression. Returns an array with the values of f evaluated for the respective solutions. Additional substitution rules can be specified in rules in the format ("a" => val) or (a => val)
source
Plotting solutions ​
The function plot is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by sort_solutions.
RecipesBase.plot Method
julia
plot(
+) -> Vector
Takes a Result object and a string f representing a Symbolics.jl expression. Returns an array with the values of f evaluated for the respective solutions. Additional substitution rules can be specified in rules in the format ("a" => val) or (a => val)
source
Plotting solutions ​
The function plot is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by sort_solutions.
RecipesBase.plot Method
julia
plot(
     res::HarmonicBalance.Result,
     varargs...;
     cut,
     kwargs...
 ) -> Plots.Plot
Plot a Result object.
Class selection done by passing String or Vector{String} as kwarg:
class       :   only plot solutions in this class(es) ("all" --> plot everything)
 not_class   :   do not plot solutions in this class(es)
Other kwargs are passed onto Plots.gr().
See also plot!
The x,y,z arguments are Strings compatible with Symbolics.jl, e.g., y=2*sqrt(u1^2+v1^2) plots the amplitude of the first quadratures multiplied by 2.
1D plots
plot(res::Result; x::String, y::String, class="default", not_class=[], kwargs...)
-plot(res::Result, y::String; kwargs...) # take x automatically from Result
Default behaviour is to plot stable solutions as full lines, unstable as dashed.
If a sweep in two parameters were done, i.e., dim(res)==2, a one dimensional cut can be plotted by using the keyword cut were it takes a Pair{Num, Float64} type entry. For example, plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2)) plots a cut at λ = 0.2.
2D plots
plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)
To make the 2d plot less chaotic it is required to specify the specific branch to plot, labeled by a Int64.
The x and y axes are taken automatically from res
source
Plotting phase diagrams ​
In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. plot_phase_diagram handles this for 1D and 2D datasets.
HarmonicBalance.plot_phase_diagram Function
julia
plot_phase_diagram(
+plot(res::Result, y::String; kwargs...) # take x automatically from Result
Default behaviour is to plot stable solutions as full lines, unstable as dashed.
If a sweep in two parameters were done, i.e., dim(res)==2, a one dimensional cut can be plotted by using the keyword cut were it takes a Pair{Num, Float64} type entry. For example, plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2)) plots a cut at λ = 0.2.
2D plots
plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)
To make the 2d plot less chaotic it is required to specify the specific branch to plot, labeled by a Int64.
The x and y axes are taken automatically from res
source
Plotting phase diagrams ​
In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. plot_phase_diagram handles this for 1D and 2D datasets.
HarmonicBalance.plot_phase_diagram Function
julia
plot_phase_diagram(
     res::HarmonicBalance.Result;
     kwargs...
 ) -> Plots.Plot
Plot the number of solutions in a Result object as a function of the parameters. Works with 1D and 2D datasets.
Class selection done by passing String or Vector{String} as kwarg:
class::String       :   only count solutions in this class ("all" --> plot everything)
-not_class::String   :   do not count solutions in this class
Other kwargs are passed onto Plots.gr()
source
Plot spaghetti plot ​
Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with plot_spaghetti.
HarmonicBalance.plot_spaghetti Function
julia
plot_spaghetti(res::Result; x, y, z, kwargs...)
Plot a three dimension line plot of a Result object as a function of the parameters. Works with 1D and 2D datasets.
Class selection done by passing String or Vector{String} as kwarg:
class::String       :   only count solutions in this class ("all" --> plot everything)
-not_class::String   :   do not count solutions in this class
Other kwargs are passed onto Plots.gr()
source

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+not_class::String   :   do not count solutions in this class
Other kwargs are passed onto Plots.gr()
source
Plot spaghetti plot ​
Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with plot_spaghetti.
HarmonicBalance.plot_spaghetti Function
julia
plot_spaghetti(res::Result; x, y, z, kwargs...)
Plot a three dimension line plot of a Result object as a function of the parameters. Works with 1D and 2D datasets.
Class selection done by passing String or Vector{String} as kwarg:
class::String       :   only count solutions in this class ("all" --> plot everything)
+not_class::String   :   do not count solutions in this class
Other kwargs are passed onto Plots.gr()
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     Saving and loading | HarmonicBalance.jl
     
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HarmonicBalance.jl
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Saving and loading ​
All of the types native to HarmonicBalance.jl can be saved into a .jld2 file using save and loaded using load. Most of the saving/loading is performed using the package JLD2.jl, with the addition of reinstating the symbolic variables in the HarmonicBalance namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as Result can be saved and loaded with no loss of information.
The function export_csv saves a .csv file which can be plot elsewhere.
HarmonicBalance.save Function
julia
save(filename, object)
Saves object into .jld2 file filename (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.
source
HarmonicBalance.load Function
julia
load(filename)
Loads an object from filename. For objects containing symbolic expressions such as HarmonicEquation, the symbolic variables are reinstated in the HarmonicBalance namespace.
source
HarmonicBalance.export_csv Function
julia
export_csv(filename, res, branch)
Saves into filename a specified solution branch of the Result res.
source

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HarmonicBalance.jl
Appearance
Saving and loading ​
All of the types native to HarmonicBalance.jl can be saved into a .jld2 file using save and loaded using load. Most of the saving/loading is performed using the package JLD2.jl, with the addition of reinstating the symbolic variables in the HarmonicBalance namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as Result can be saved and loaded with no loss of information.
The function export_csv saves a .csv file which can be plot elsewhere.
HarmonicBalance.save Function
julia
save(filename, object)
Saves object into .jld2 file filename (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.
source
HarmonicBalance.load Function
julia
load(filename)
Loads an object from filename. For objects containing symbolic expressions such as HarmonicEquation, the symbolic variables are reinstated in the HarmonicBalance namespace.
source
HarmonicBalance.export_csv Function
julia
export_csv(filename, res, branch)
Saves into filename a specified solution branch of the Result res.
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diff --git a/dev/manual/solving_harmonics.html b/dev/manual/solving_harmonics.html
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     Solving harmonic equations | HarmonicBalance.jl
     
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HarmonicBalance.jl
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Solving harmonic equations ​
Once a differential equation of motion has been defined in DifferentialEquation and converted to a HarmonicEquation, we may use the homotopy continuation method (as implemented in HomotopyContinuation.jl) to find steady states. This means that, having called get_harmonic_equations, we need to set all time-derivatives to zero and parse the resulting algebraic equations into a Problem.
Problem holds the steady-state equations, and (optionally) the symbolic Jacobian which is needed for stability / linear response calculations.
Once defined, a Problem can be solved for a set of input parameters using get_steady_states to obtain Result.
HarmonicBalance.Problem Type
julia
mutable struct Problem
Holds a set of algebraic equations describing the steady state of a system.
Fields
  • variables::Vector{Num}: The harmonic variables to be solved for.
  • parameters::Vector{Num}: All symbols which are not the harmonic variables.
  • system::HomotopyContinuation.ModelKit.System: The input object for HomotopyContinuation.jl solver methods.
  • jacobian::Any: The Jacobian matrix (possibly symbolic). If false, the Jacobian is ignored (may be calculated implicitly after solving).
  • eom::HarmonicEquation: The HarmonicEquation object used to generate this Problem.
Constructors
julia
Problem(eom::HarmonicEquation; Jacobian=true) # find and store the symbolic Jacobian
+    
HarmonicBalance.jl
Appearance
Solving harmonic equations ​
Once a differential equation of motion has been defined in DifferentialEquation and converted to a HarmonicEquation, we may use the homotopy continuation method (as implemented in HomotopyContinuation.jl) to find steady states. This means that, having called get_harmonic_equations, we need to set all time-derivatives to zero and parse the resulting algebraic equations into a Problem.
Problem holds the steady-state equations, and (optionally) the symbolic Jacobian which is needed for stability / linear response calculations.
Once defined, a Problem can be solved for a set of input parameters using get_steady_states to obtain Result.
HarmonicBalance.Problem Type
julia
mutable struct Problem
Holds a set of algebraic equations describing the steady state of a system.
Fields
  • variables::Vector{Num}: The harmonic variables to be solved for.
  • parameters::Vector{Num}: All symbols which are not the harmonic variables.
  • system::HomotopyContinuation.ModelKit.System: The input object for HomotopyContinuation.jl solver methods.
  • jacobian::Any: The Jacobian matrix (possibly symbolic). If false, the Jacobian is ignored (may be calculated implicitly after solving).
  • eom::HarmonicEquation: The HarmonicEquation object used to generate this Problem.
Constructors
julia
Problem(eom::HarmonicEquation; Jacobian=true) # find and store the symbolic Jacobian
 Problem(eom::HarmonicEquation; Jacobian="implicit") # ignore the Jacobian for now, compute implicitly later
 Problem(eom::HarmonicEquation; Jacobian=J) # use J as the Jacobian (a function that takes a Dict)
-Problem(eom::HarmonicEquation; Jacobian=false) # ignore the Jacobian
source
HarmonicBalance.get_steady_states Function
julia
get_steady_states(problem::HarmonicEquation,
+Problem(eom::HarmonicEquation; Jacobian=false) # ignore the Jacobian
source
HarmonicBalance.get_steady_states Function
julia
get_steady_states(problem::HarmonicEquation,
                     method::HarmonicBalanceMethod,
                     swept_parameters::ParameterRange,
                     fixed_parameters::ParameterList;
@@ -53,7 +54,7 @@
        of which real:    1
        of which stable:  1
 
-    Classes: stable, physical, Hopf, binary_labels
source
HarmonicBalance.Result Type
julia
mutable struct Result
Stores the steady states of a HarmonicEquation.
Fields
  • solutions::Array{Vector{Vector{ComplexF64}}}: The variable values of steady-state solutions.
  • swept_parameters::OrderedCollections.OrderedDict{Num, Vector{Union{Float64, ComplexF64}}}: Values of all parameters for all solutions.
  • fixed_parameters::OrderedCollections.OrderedDict{Num, Float64}: The parameters fixed throughout the solutions.
  • problem::HarmonicBalance.Problem: The Problem used to generate this.
  • classes::Dict{String, Array}: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method classify_solutions!).
  • jacobian::Function: The Jacobian with fixed_parameters already substituted. Accepts a dictionary specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function. If problem.jacobian was false, this holds a function that rearranges the equations to find J only after numerical values are inserted (preferable in cases where the symbolic J would be very large).
  • seed::UInt32: Seed used for the solver
source
Classifying solutions ​
The solutions in Result are accompanied by similarly-sized boolean arrays stored in the dictionary Result.classes. The classes can be used by the plotting functions to show/hide/label certain solutions.
By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with classify_solutions!.
HarmonicBalance.classify_solutions! Function
julia
classify_solutions!(
+    Classes: stable, physical, Hopf, binary_labels
source
HarmonicBalance.Result Type
julia
mutable struct Result
Stores the steady states of a HarmonicEquation.
Fields
  • solutions::Array{Vector{Vector{ComplexF64}}}: The variable values of steady-state solutions.
  • swept_parameters::OrderedCollections.OrderedDict{Num, Vector{Union{Float64, ComplexF64}}}: Values of all parameters for all solutions.
  • fixed_parameters::OrderedCollections.OrderedDict{Num, Float64}: The parameters fixed throughout the solutions.
  • problem::HarmonicBalance.Problem: The Problem used to generate this.
  • classes::Dict{String, Array}: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method classify_solutions!).
  • jacobian::Function: The Jacobian with fixed_parameters already substituted. Accepts a dictionary specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function. If problem.jacobian was false, this holds a function that rearranges the equations to find J only after numerical values are inserted (preferable in cases where the symbolic J would be very large).
  • seed::UInt32: Seed used for the solver
source
Classifying solutions ​
The solutions in Result are accompanied by similarly-sized boolean arrays stored in the dictionary Result.classes. The classes can be used by the plotting functions to show/hide/label certain solutions.
By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with classify_solutions!.
HarmonicBalance.classify_solutions! Function
julia
classify_solutions!(
     res::HarmonicBalance.Result,
     func::Union{Function, String},
     name::String;
@@ -62,12 +63,12 @@
 res = get_steady_states(problem, swept_parameters, fixed_parameters)
 
 # classify, store in result.classes["large_amplitude"]
-classify_solutions!(res, "sqrt(u1^2 + v1^2) > 1.0" , "large_amplitude")
source
Sorting solutions ​
Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.
For stable states, the branches describe a system's behaviour under adiabatic parameter changes.
Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.
The function sort_solutions goes over the the raw output of get_steady_states and sorts each entry such that neighboring solution sets minimize Euclidean distance.
Currently, sort_solutions is compatible with 1D and 2D arrays of solution sets.
HarmonicBalance.sort_solutions Function
julia
sort_solutions(
+classify_solutions!(res, "sqrt(u1^2 + v1^2) > 1.0" , "large_amplitude")
source
Sorting solutions ​
Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.
For stable states, the branches describe a system's behaviour under adiabatic parameter changes.
Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.
The function sort_solutions goes over the the raw output of get_steady_states and sorts each entry such that neighboring solution sets minimize Euclidean distance.
Currently, sort_solutions is compatible with 1D and 2D arrays of solution sets.
HarmonicBalance.sort_solutions Function
julia
sort_solutions(
     solutions::Array;
     sorting,
     show_progress
-) -> Array
Sorts solutions into branches according to the method sorting.
solutions is an n-dimensional array of Vector{Vector}. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.
Keyword arguments
  • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".
  • show_progress: Indicate whether a progress bar should be displayed.
source

-    
+) -> Array
Sorts solutions into branches according to the method sorting.
solutions is an n-dimensional array of Vector{Vector}. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.
Keyword arguments
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     Time evolution | HarmonicBalance.jl
     
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HarmonicBalance.jl
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Time evolution ​
Generally, solving the ODE of oscillatory systems in time requires numerically tracking the oscillations. This is a computationally expensive process; however, using the harmonic ansatz removes the oscillatory time-dependence. Simulating instead the harmonic variables of a HarmonicEquation is vastly more efficient - a steady state of the system appears as a fixed point in multidimensional space rather than an oscillatory function.
The Extention TimeEvolution is used to interface HarmonicEquation with the solvers contained in OrdinaryDiffEq.jl. Time-dependent parameter sweeps are defined using the object AdiabaticSweep. To use the TimeEvolution extension, one must first load the OrdinaryDiffEq.jl package.
SciMLBase.ODEProblem Method
julia
ODEProblem(
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HarmonicBalance.jl
Appearance
Time evolution ​
Generally, solving the ODE of oscillatory systems in time requires numerically tracking the oscillations. This is a computationally expensive process; however, using the harmonic ansatz removes the oscillatory time-dependence. Simulating instead the harmonic variables of a HarmonicEquation is vastly more efficient - a steady state of the system appears as a fixed point in multidimensional space rather than an oscillatory function.
The Extention TimeEvolution is used to interface HarmonicEquation with the solvers contained in OrdinaryDiffEq.jl. Time-dependent parameter sweeps are defined using the object AdiabaticSweep. To use the TimeEvolution extension, one must first load the OrdinaryDiffEq.jl package.
SciMLBase.ODEProblem Method
julia
ODEProblem(
         eom::HarmonicEquation;
         fixed_parameters,
         u0::Vector,
         sweep::AdiabaticSweep,
         timespan::Tuple
-        )
Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in eom to simulate time-evolution within timespan. fixed_parameters must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If u0 is specified, it is used as an initial condition; otherwise the values from fixed_parameters are used.
source
HarmonicBalance.AdiabaticSweep Type
Represents a sweep of one or more parameters of a HarmonicEquation. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.
Sweeps of different variables can be combined using +.
Fields
  • functions::Dict{Num, Function}: Maps each swept parameter to a function.
Examples
julia
# create a sweep of parameter a from 0 to 1 over time 0 -> 100
+        )
Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in eom to simulate time-evolution within timespan. fixed_parameters must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If u0 is specified, it is used as an initial condition; otherwise the values from fixed_parameters are used.
source
HarmonicBalance.AdiabaticSweep Type
Represents a sweep of one or more parameters of a HarmonicEquation. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.
Sweeps of different variables can be combined using +.
Fields
  • functions::Dict{Num, Function}: Maps each swept parameter to a function.
Examples
julia
# create a sweep of parameter a from 0 to 1 over time 0 -> 100
 julia> @variables a,b;
 julia> sweep = AdiabaticSweep(a => [0., 1.], (0, 100));
 julia> sweep[a](50)
@@ -39,18 +40,18 @@
 julia> sweep = AdiabaticSweep([a => [0.,1.], b => [0., 1.]], (0,100))
Successive sweeps can be combined,
julia
sweep1 = AdiabaticSweep(ω => [0.95, 1.0], (0, 2e4))
 sweep2 = AdiabaticSweep(λ => [0.05, 0.01], (2e4, 4e4))
 sweep = sweep1 + sweep2
multiple parameters can be swept simultaneously,
julia
sweep = AdiabaticSweep([ω => [0.95;1.0], λ => [5e-2;1e-2]], (0, 2e4))
and custom sweep functions may be used.
julia
ωfunc(t) = cos(t)
-sweep = AdiabaticSweep(ω => ωfunc)
source
Plotting ​
RecipesBase.plot Method
julia
plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)
Plot a function f of a time-dependent solution soln of harm_eq.
As a function of time
plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)
f is parsed by Symbolics.jl
parametric plots
plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)
Parametric plot of f[1] against f[2]
Also callable as plot!
source
Miscellaneous ​
Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.
HarmonicBalance.is_stable Function
julia
is_stable(
+sweep = AdiabaticSweep(ω => ωfunc)
source
Plotting ​
RecipesBase.plot Method
julia
plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)
Plot a function f of a time-dependent solution soln of harm_eq.
As a function of time
plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)
f is parsed by Symbolics.jl
parametric plots
plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)
Parametric plot of f[1] against f[2]
Also callable as plot!
source
Miscellaneous ​
Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.
HarmonicBalance.is_stable Function
julia
is_stable(
     soln::OrderedCollections.OrderedDict{Num, ComplexF64},
     eom::HarmonicEquation;
     timespan,
     tol,
     perturb_initial
-)
Numerically investigate the stability of a solution soln of eom within timespan. The initial condition is displaced by perturb_initial.
Return true the solution evolves within tol of the initial value (interpreted as stable).
source
julia
is_stable(
+)
Numerically investigate the stability of a solution soln of eom within timespan. The initial condition is displaced by perturb_initial.
Return true the solution evolves within tol of the initial value (interpreted as stable).
source
julia
is_stable(
     soln::OrderedCollections.OrderedDict{Num, ComplexF64},
     res::HarmonicBalance.Result;
     kwargs...
-) -> Any
Returns true if the solution soln of the Result res is stable. Stable solutions are real and have all Jacobian eigenvalues Re[λ] <= 0. im_tol : an absolute threshold to distinguish real/complex numbers. rel_tol: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)
source

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+) -> Any
Returns true if the solution soln of the Result res is stable. Stable solutions are real and have all Jacobian eigenvalues Re[λ] <= 0. im_tol : an absolute threshold to distinguish real/complex numbers. rel_tol: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)
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diff --git a/dev/tutorials/classification.html b/dev/tutorials/classification.html
index efbb9145..e2b4791b 100644
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     Classifying solutions | HarmonicBalance.jl
     
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HarmonicBalance.jl
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Classifying solutions ​
Given that you obtained some steady states for a parameter sweep of a specific model it can be useful to classify these solution. Let us consider a simple pametric oscillator
julia
using HarmonicBalance
+    
HarmonicBalance.jl
Appearance
Classifying solutions ​
Given that you obtained some steady states for a parameter sweep of a specific model it can be useful to classify these solution. Let us consider a simple pametric oscillator
julia
using HarmonicBalance
 
 @variables ω₀ γ λ α ω t x(t)
 
@@ -66,7 +67,7 @@
 
 Classes: zero, stable, physical, Hopf, binary_labels
We can visualize the zero amplitude solution:
julia
plot_phase_diagram(result_2D, class=["zero", "stable"])
This shows that inside the Mathieu lobe the zero amplitude solution becomes unstable due to the parametric drive being resonant with the oscillator.
We can also visualize the equi-amplitude curves of the solutions:
julia
classify_solutions!(result_2D, "sqrt(u1^2 + v1^2) > 0.12", "large amplitude")
 plot_phase_diagram(result_2D, class=["large amplitude", "stable"])

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diff --git a/dev/tutorials/index.html b/dev/tutorials/index.html
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     Tutorials | HarmonicBalance.jl
     
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diff --git a/dev/tutorials/limit_cycles.html b/dev/tutorials/limit_cycles.html
index 9db90c51..61398743 100644
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     Limit cycles | HarmonicBalance.jl
     
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HarmonicBalance.jl
Appearance
Limit cycles ​
In contrast to the previous tutorials, limit cycle problems feature harmonic(s) whose numerical value is not imposed externally. We shall construct our HarmonicEquation as usual, but identify this harmonic as an extra variable, rather than a fixed parameter.
Non-driven system - the van der Pol oscillator ​
Here we solve the equation of motion of the van der Pol oscillator. This is a single-variable second-order ODE with continuous time-translation symmetry (i.e., no 'clock' imposing a frequency and/or phase), which displays periodic solutions known as relaxation oscillations. For more detail, refer also to arXiv:2308.06092.
julia
using HarmonicBalance
+    
HarmonicBalance.jl
Appearance
Limit cycles ​
In contrast to the previous tutorials, limit cycle problems feature harmonic(s) whose numerical value is not imposed externally. We shall construct our HarmonicEquation as usual, but identify this harmonic as an extra variable, rather than a fixed parameter.
Non-driven system - the van der Pol oscillator ​
Here we solve the equation of motion of the van der Pol oscillator. This is a single-variable second-order ODE with continuous time-translation symmetry (i.e., no 'clock' imposing a frequency and/or phase), which displays periodic solutions known as relaxation oscillations. For more detail, refer also to arXiv:2308.06092.
julia
using HarmonicBalance
 @variables ω_lc, t, ω0, x(t), μ
 diff_eq = DifferentialEquation(d(d(x,t),t) - μ*(1-x^2) * d(x,t) + x, x)
System of 1 differential equations
 Variables:       x(t)
@@ -111,7 +112,7 @@
 time_evo = solve(time_problem, Tsit5(), saveat=100);
Inspecting the amplitude as a function of time,
julia
plot(time_evo, "sqrt(u1^2 + v1^2)", harmonic_eq)
we see that initially the sweep is adiabatic as it proceeds along the steady-state branch 1. At around T=2e6, an instability occurs and u1(T) starts to rapidly oscillate. At that point, the sweep is stopped. Under free time evolution, the system then settles into a limit-cycle solution where the coordinates move along closed trajectories.
By plotting the u and v variables against each other, we observe the limit cycle shapes in phase space,
julia
p1 = plot(time_evo, ["u1", "v1"], harmonic_eq)
 p2 = plot(time_evo, ["u2", "v2"], harmonic_eq)
 plot(p1, p2)

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diff --git a/dev/tutorials/linear_response.html b/dev/tutorials/linear_response.html
index e7e1f0da..846c42e2 100644
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     Linear response | HarmonicBalance.jl
     
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HarmonicBalance.jl
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Linear response ​
In HarmonicBalance.jl, the stability and linear response are treated using the LinearResponse module.
Here we calculate the white noise response of a simple nonlinear system. A set of reference results may be found in Huber et al. in Phys. Rev. X 10, 021066 (2020). We start by defining the Duffing oscillator
julia
using HarmonicBalance, Plots
+    
HarmonicBalance.jl
Appearance
Linear response ​
In HarmonicBalance.jl, the stability and linear response are treated using the LinearResponse module.
Here we calculate the white noise response of a simple nonlinear system. A set of reference results may be found in Huber et al. in Phys. Rev. X 10, 021066 (2020). We start by defining the Duffing oscillator
julia
using HarmonicBalance, Plots
 using Plots.Measures: mm
 @variables α, ω, ω0, F, γ, t, x(t); # declare constant variables and a function x(t)
 
@@ -72,7 +73,7 @@
   plot_linear_response(result, x, branch=1, Ω_range=range(0.9,1.1,300), logscale=true, xscale=:log),
   size=(600, 250), margin=3mm
 )
We see that for low F, quasi-linear behaviour with a single Lorentzian response occurs, while for larger F, two peaks form in the noise response. The two peaks are strongly unequal in magnitude, which is an example of internal squeezing (See supplemental material of Huber et al.).

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diff --git a/dev/tutorials/steady_states.html b/dev/tutorials/steady_states.html
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     Finding the staedy states of a Duffing oscillator | HarmonicBalance.jl
     
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HarmonicBalance.jl
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Finding the staedy states of a Duffing oscillator ​
Here we show the workflow of HarmonicBalance.jl on a simple example - the driven Duffing oscillator. The equation of motion for the displacement x(t) reads
x¨(t)+γx˙(t)+ω02x(t)⏟damped harmonic oscillator+αx(t)3⏟Duffing coefficient=Fcos⁡(ωt)⏟periodic drive
In general, there is no analytical solution to the differential equation. Fortunately, some harmonics are more important than others. By truncating the infinite-dimensional Fourier space to a set of judiciously chosen harmonics, we may obtain a soluble system. For the Duffing resonator, we can well try to only consider the drive frequency ω. To implement this, we use the harmonic ansatz
x(t)=Ucos⁡(ωt)+Vsin⁡(ωt),
which constraints the spectrum of x(t) to a single harmonic. Fixing the quadratures U and V to be constant then reduces the differential equation to two coupled cubic polynomial equations (for more details on this step, see the appendices in the white paper). Finding the roots of coupled polynomials is in general very hard. We here apply the method of homotopy continuation, as implemented in HomotopyContinuation.jl which is guaranteed to find the complete set of roots.
First we need to declare the symbolic variables (the excellent Symbolics.jl is used here).
julia
using HarmonicBalance
+    
HarmonicBalance.jl
Appearance
Finding the staedy states of a Duffing oscillator ​
Here we show the workflow of HarmonicBalance.jl on a simple example - the driven Duffing oscillator. The equation of motion for the displacement x(t) reads
x¨(t)+γx˙(t)+ω02x(t)⏟damped harmonic oscillator+αx(t)3⏟Duffing coefficient=Fcos⁡(ωt)⏟periodic drive
In general, there is no analytical solution to the differential equation. Fortunately, some harmonics are more important than others. By truncating the infinite-dimensional Fourier space to a set of judiciously chosen harmonics, we may obtain a soluble system. For the Duffing resonator, we can well try to only consider the drive frequency ω. To implement this, we use the harmonic ansatz
x(t)=Ucos⁡(ωt)+Vsin⁡(ωt),
which constraints the spectrum of x(t) to a single harmonic. Fixing the quadratures U and V to be constant then reduces the differential equation to two coupled cubic polynomial equations (for more details on this step, see the appendices in the white paper). Finding the roots of coupled polynomials is in general very hard. We here apply the method of homotopy continuation, as implemented in HomotopyContinuation.jl which is guaranteed to find the complete set of roots.
First we need to declare the symbolic variables (the excellent Symbolics.jl is used here).
julia
using HarmonicBalance
 @variables α ω ω0 F γ t x(t) # declare constant variables and a function x(t)
Next, we have to input the equations of motion. This will be stored as a DifferentialEquation. The input needs to specify that only x is a mathematical variable, the other symbols are parameters:
julia
diff_eq = DifferentialEquation(d(x,t,2) + ω0^2*x + α*x^3 + γ*d(x,t) ~ F*cos(ω*t), x)
System of 1 differential equations
 Variables:       x(t)
 Harmonic ansatz: x(t) => ;   
@@ -73,7 +74,7 @@
 Classes: stable, physical, Hopf, binary_labels
Although 9 branches were found in total, only 3 remain physical (real-valued). Let us visualise the amplitudes corresponding to the two harmonics, U12+V12 and U22+V22 :
julia
p1 = plot(result, "sqrt(u1^2 + v1^2)", legend=false)
 p2 = plot(result, "sqrt(u2^2 + v2^2)")
 plot(p1, p2)
The contributions of ω and 3ω are now comparable and the system shows some fairly complex behaviour! This demonstrates how an exact solution within an extended Fourier subspace goes beyond a perturbative treatment.

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diff --git a/dev/tutorials/time_dependent.html b/dev/tutorials/time_dependent.html
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     Time-dependent simulations | HarmonicBalance.jl
     
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HarmonicBalance.jl
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Time-dependent simulations ​
Most of HarmonicBalance.jl is focused on finding and analysing the steady states. Such states contain no information about transient behaviour, which is crucial to answer the following.
  • Given an initial condition, which steady state does the system evolve into?
  • How does the system behave if its parameters are varied in time?
It is straightforward to evolve the full equation of motion using an ODE solver. However, tracking oscillatory behaviour is computationally expensive.
In the background, we showed that nonlinear driven systems may be reduced to harmonic equations
du(T)dT=F¯(u),
As long as the chosen harmonics constituting u(T) capture the system's behaviour, we may numerically evolve the new effective differential equations instead of the full problem. Since the components of u(T) only vary very slowly (and are constant in a steady state), this is usually vastly more efficient than evolving the full problem.
Here we primarily demonstrate on the parametrically driven oscillator.
We start by defining our system.
julia
using HarmonicBalance
+    
HarmonicBalance.jl
Appearance
Time-dependent simulations ​
Most of HarmonicBalance.jl is focused on finding and analysing the steady states. Such states contain no information about transient behaviour, which is crucial to answer the following.
  • Given an initial condition, which steady state does the system evolve into?
  • How does the system behave if its parameters are varied in time?
It is straightforward to evolve the full equation of motion using an ODE solver. However, tracking oscillatory behaviour is computationally expensive.
In the background, we showed that nonlinear driven systems may be reduced to harmonic equations
du(T)dT=F¯(u),
As long as the chosen harmonics constituting u(T) capture the system's behaviour, we may numerically evolve the new effective differential equations instead of the full problem. Since the components of u(T) only vary very slowly (and are constant in a steady state), this is usually vastly more efficient than evolving the full problem.
Here we primarily demonstrate on the parametrically driven oscillator.
We start by defining our system.
julia
using HarmonicBalance
 @variables ω0 γ λ F θ η α ω t x(t)
 
 eq =  d(d(x,t),t) + γ*d(x,t) + ω0^2*(1 - λ*cos(2*ω*t))*x + α*x^3 + η*d(x,t)*x^2 ~ F*cos(ω*t + θ)
@@ -58,7 +59,7 @@
 plot(result, "sqrt(u1^2 + v1^2)")
Clearly when evolving from u0 = [0., 0.], the system ends up in the low-amplitude branch 2. With u0 = [0.2, 0.2], the system ends up in branch 3.
Adiabatic parameter sweeps ​
Experimentally, the primary means of exploring the steady state landscape is an adiabatic sweep one or more of the system parameters. This takes the system along a solution branch. If this branch disappears or becomes unstable, a jump occurs.
The object AdiabaticSweep specifies a sweep, which is then used as an optional sweep keyword in the ODEProblem constructor.
julia
sweep = AdiabaticSweep(ω => (0.9,1.1), (0, 2e4))
AdiabaticSweep(Dict{Num, Function}(ω => TimeEvolution.var"#f#1"{Tuple{Float64, Float64}, Float64, Int64}((0.9, 1.1), 20000.0, 0)))
The sweep linearly interpolates between ω=0.9 at time 0 and ω=1.1 at time 2e4. For earlier/later times, ω is constant.
Let us now define a new ODEProblem which incorporates sweep and again use solve:
julia
ode_problem = ODEProblem(harmonic_eq, fixed, sweep=sweep, u0=[0.1;0.0], timespan=(0, 2e4))
 time_evo = solve(ode_problem, Tsit5(), saveat=100)
 plot(time_evo, "sqrt(u1^2 + v1^2)", harmonic_eq)
We see the system first evolves from the initial condition towards the low-amplitude steady state. The amplitude increases as the sweep proceeds, with a jump occurring around ω=1.08 (i.e., time 18000).

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