Balthasar Van Der Pols Second Order Differential Equation Worksheet

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dy
Dutch electrical engineering Balthasar van der Pol originally derived a second-order differential
equation:
dy
= u(1 – y2)
у
dt2
dt
to model the behavior of an electrical circuit with vacuum tubes (1927,
https://en.wikipedia.org/wiki/Van der Pol oscillator). The equation was later adopted by Fitzhugh and
Nagumo to describe action potential movement through neurons (i.e., the signal that moves through
nerves from your fingertip to your brain when you touch something hot).
You have been hired by the National Institutes of Health (NIH) to solve this equation and examine how
the solution changes for different values of u. The NIH recommends that you introduce a new variable,
Yı, and use a standard transformation to change the equation into a system of two first-order
equations:
dy
= y1
dt
dyı
u(1 – y)yı – y
dt
and use the initial conditions y(t = 0) = 1.0 and yı(t = 0) = 1.0.
=
The NIH would like to compare (and plot) the action potential, y, behavior for different values of u
between 100 and 800. You should simulate t = 0 to 1000 (dimensionless). The value of u is known to
change as humans age and it is known to vary from person to person so understanding the impact of u
on nerve signaling is important. Further, the equations become more difficult to solve (the solution
becomes unstable) as u increases, so you should use scipy.integrate.odeint() and smaller time steps may
be required.
Note: Plotting the potential, y, on the x-axis versus the rate of change of the potential, Yı, on the y-axis
shows interesting oscillatory behavior (see figure below for u = 100) that gets ‘sharper’ as u increases.
100
50
Yı, rate of change
0
-50
-100
10
15
-2.0 -1.5 -1.0 -0.5 0.0 0.5
Potential
20