Van der Pol’s differential equation is The equation describes a system with nonlinear damping, the degree of nonlinearity given by μ.

If μ = 0 the system is linear and undamped, but as μ increases the strength of the nonlinearity increases.

We will plot the phase portrait for the solution to Van der Pol’s equation in Python using SciPy’s new ODE solver ivp_solve.

The function ivp_solve does not solve second-order systems of equations directly.

It solves systems of first-order equations, but a second-order differential equation can be recast as a pair of first-order equations by introducing the first derivative as a new variable.

Since y is the derivative of x, the phase portrait is just the plot of (x, y).

If μ = 0, we have a simple harmonic oscillator and the phase portrait is simply a circle.

For larger values of μ the solutions enter limiting cycles, but the cycles are more complicated than just circles.

Here’s the Python code that made the plot.

from scipy import linspace from scipy.

integrate import solve_ivp import matplotlib.

pyplot as plt def vdp(t, z): x, y = z return [y, mu*(1 – x**2)*y – x] a, b = 0, 10 mus = [0, 1, 2] styles = [“-“, “–“, “:”] t = linspace(a, b, 500) for mu, style in zip(mus, styles): sol = solve_ivp(vdp, [a, b], [1, 0], t_eval=t) plt.

plot(sol.

y[0], sol.

y[1], style) # make a little extra horizontal room for legend plt.

xlim([-3,3]) plt.

legend([f”$mu={m}$” for m in mus]) plt.

axes().

set_aspect(1) To plot the solutions as a function of time, rather than plotting phase portraits, change the line plt.

plot(sol.

y[0], sol.

y[1], style) to plt.

plot(sol.

t, sol.

y[0], style) and comment out the line setting xlim This gives the following plot.

Related posts Duffing equation The other butterfly effect Linear mechanical vibrations.