Figures2_and_3.nb

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Generating Figures 2 and 3 in the article \[OpenCurlyDoubleQuote]Obscuring \
digital route choice information prevents delay-induced congestion\
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Figure 2: Routing in a two-route network with delayed information\
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Solving the delay-differential equations by numerical integration\
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With this notebook, we solve the delayed differential equations of vehicle \
numbers on two streets numerically by integration, plot some resulting street \
load dynamics and find whether congestion arises for given sets of parameters.

To start, we first define our travel time, the corresponding in-rates and the \
differential equations for the two streets A and C.\
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Our differential equation has two fixed points, which we find by determining \
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Now we have all the tools to integrate the differential equations numerically \
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In Figs. 2b-d of the article, we show the numerical DDE solutions for three \
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In addition to these exemplary dynamics, we want to determine for which pairs \
of in-rate and delay congestion occurs. To this end, we define a function \
which for each DDE solution finds whether the overall number of vehicles on \
the streets continuously rises or not.
We save the results for the considered range of delays and in-rates as a \
csv-file to then plot the phase diagram with python (see the jupyter notebook \
and respective modules for details).\
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In addition to the numerical integration of the DDE, we also determine the \
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The general idea is to linearize the differential equation around its stable \
fixed point and then for each delay find a critical in-rate with an \
eigenvalue with zero real part.

So again, we formulate our differential equations and from them derive two \
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*)