Van der Pol Oscillator

2D Ordinary Differential Equations

The Van der Pol oscillator is a prototypical nonlinear system that exhibits self-sustained oscillations. For a wide range of initial conditions, trajectories converge to a stable limit cycle.

In the form we will use in this course, the system is written as (Strogatz 2024, chap. 7.5):

\[\begin{aligned} \dot x &= \mu\,(y - f(x)) \\ \dot y &= -\frac{x}{\mu} \end{aligned}\]

where $\mu>0$ controls the strength of the nonlinearity (and the fast–slow character for large $\mu$), and

\[ f(x) = \frac{x^3}{3} - x. \]

Figure 1: Phase plane of the Van der Pol system with mu = 3. Trajectories are shown for different initial conditions (IC as x-y, marked black). Generated with `scipy.integrate.solve_ivp()` and matplotlib.

Follow the steps outline in the CDIMA model to explore how the Van der Pol system behaves for different initial conditions and parameters.

The Stable Limit Cycle

The Van der Pol system has a stable limit cycle that attracts trajectories from a wide range of initial conditions. Plot this limit cycle as a $x(t)-t$ plot, and verify that it matches the phase plane trajectories.

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Figure 2: Time series of x(t) for the Van der Pol system with mu = 3 and IC: (2, -2). Generated with `scipy.integrate.solve_ivp()` and matplotlib.

The amplitude and period of the limit cycle depend on $\mu$. There are two phases of the limit cycle: a slow phase where trajectories move slowly near the cubic nullcline, and a fast phase where trajectories rapidly jump between branches of the nullcline. The speed of the slow phase increases with $\mu$, while the fast phase becomes more abrupt. This leads to a characteristic “relaxation oscillation” shape for large $\mu$.

Play with the parameter $\mu$ to see how it changes the shape of the limit cycle and the transient dynamics.

Resources

(Strogatz 2024, chap. 7.5) for theoretical background on the Van der Pol oscillator.
SciPy documentation: https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html
Reference implementation: https://github.com/daniprec/BAM-Applied-Math-Lab/tree/main/sessions/s02_odes_2d

References

Strogatz, Steven H. 2024. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Chapman; Hall/CRC.

--- title: "Van der Pol Oscillator" subtitle: "2D Ordinary Differential Equations" format: html --- The Van der Pol oscillator is a prototypical nonlinear system that exhibits **self-sustained oscillations**. For a wide range of initial conditions, trajectories converge to a stable **limit cycle**. In the form we will use in this course, the system is written as [@strogatz2024nonlinear, chap. 7.5]: $$\begin{aligned} \dot x &= \mu\,(y - f(x)) \\ \dot y &= -\frac{x}{\mu} \end{aligned}$$ where $\mu>0$ controls the strength of the nonlinearity (and the fast–slow character for large $\mu$), and $$ f(x) = \frac{x^3}{3} - x. $$ ```{python} #| label: fig-phase-plane #| fig-cap: "Phase plane of the Van der Pol system with mu = 3. Trajectories are shown for different initial conditions (IC as x-y, marked black). Generated with `scipy.integrate.solve_ivp()` and matplotlib." #| fig-width: 8 #| fig-height: 4 #| echo: false import matplotlib.pyplot as plt import numpy as np from scipy.integrate import solve_ivp from sessions.s02_odes_2d.van_der_pol import van_der_pol ls_xy = [(-2, 2), (2, -2), (-0.1, 0.1)] mu = 3 fig, ax = plt.subplots(figsize=(8, 4)) for x0, y0 in ls_xy: sol = solve_ivp( van_der_pol, [0, 40], [x0, y0], args=(mu,), t_eval=np.linspace(0, 40, 4000), ) ax.plot(sol.y[0], sol.y[1], label=f"IC: ({x0}, {y0})") # Mark the initial condition ax.plot(x0, y0, "o", color="black") plt.xlabel("x") plt.ylabel("y") plt.legend() plt.show() plt.close() ``` Follow the steps outline in [the CDIMA model](cdima.qmd) to explore how the Van der Pol system behaves for different initial conditions and parameters. ## The Stable Limit Cycle The Van der Pol system has a stable limit cycle that attracts trajectories from a wide range of initial conditions. Plot this limit cycle as a $x(t)-t$ plot, and verify that it matches the phase plane trajectories. ```{python} #| label: fig-limit-cycle #| fig-cap: "Time series of x(t) for the Van der Pol system with mu = 3 and IC: (2, -2). Generated with `scipy.integrate.solve_ivp()` and matplotlib." #| fig-width: 8 #| fig-height: 4 #| echo: false xy = (2, -2) fig, ax = plt.subplots(figsize=(8, 4)) sol = solve_ivp( van_der_pol, [0, 40], xy, args=(mu,), t_eval=np.linspace(0, 40, 4000), ) ax.plot(sol.t, sol.y[0]) plt.xlabel("Time (t)") plt.ylabel("x(t)") plt.legend() plt.show() plt.close() ``` The amplitude and period of the limit cycle depend on $\mu$. There are two phases of the limit cycle: a slow phase where trajectories move slowly near the cubic nullcline, and a fast phase where trajectories rapidly jump between branches of the nullcline. The speed of the slow phase increases with $\mu$, while the fast phase becomes more abrupt. This leads to a characteristic "relaxation oscillation" shape for large $\mu$. Play with the parameter $\mu$ to see how it changes the shape of the limit cycle and the transient dynamics. ## Resources - [@strogatz2024nonlinear, chap. 7.5] for theoretical background on the Van der Pol oscillator. - SciPy documentation: <https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html> - Reference implementation: <https://github.com/daniprec/BAM-Applied-Math-Lab/tree/main/sessions/s02_odes_2d>