Van der Pol Oscillator

Limit Cycles and Relaxation Oscillations

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):

\[\begin{aligned} \dot x &= \mu\,(y - f(x)) \\ \dot y &= -\frac{x}{\mu} \end{aligned} \tag{1}\]

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

\[ f(x) = \frac{x^3}{3} - x. \tag{2}\]

Nullclines and Fixed Point

The nullclines are especially simple in this model:

\[ \dot x = 0 \quad \Longrightarrow \quad y = \frac{x^3}{3} - x, \qquad \dot y = 0 \quad \Longrightarrow \quad x = 0. \]

They intersect at the origin, so $(0,0)$ is the equilibrium point. For the parameter regime used in this session, the main long-run object of interest is not the equilibrium itself, but the attracting limit cycle around it.

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.

Implement the Vector Field

def van_der_pol(t, xy, mu=1.0):
    x, y = xy
    fx = x**3 / 3 - x
    dxdt = mu * (y - fx)
    dydt = -x / mu
    return np.array([dxdt, dydt])

Follow the same pipeline as in CDIMA Reaction: define the right-hand side, solve the initial value problem, compute nullclines, and then animate the resulting trajectory.

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.

/tmp/ipykernel_3383/2854713749.py:14: UserWarning:

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

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 Role of $\mu$

The amplitude and period of the limit cycle depend on $\mu$.

For smaller $\mu$, the oscillation looks smoother and less sharply separated into time scales.
For larger $\mu$, the motion develops a clear slow-fast structure.
The trajectory spends long intervals drifting near the cubic nullcline, then jumps rapidly between branches.

This is the characteristic relaxation-oscillation picture.

What to Look For

Different initial conditions approach the same closed orbit.
The transient depends on the initial state, but the long-run behavior does not.
The time trace $x(t)$ reflects the same slow-fast pattern you see in the phase plane.

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

What’s Next?

Once the geometry is clear, the next step is to animate the trajectory and, if needed, make the initial condition interactive.

Animation Template Assignment

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: "Limit Cycles and Relaxation Oscillations" 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]: $$\begin{aligned} \dot x &= \mu\,(y - f(x)) \\ \dot y &= -\frac{x}{\mu} \end{aligned}$$ {#eq-ode-2d-van-der-pol-1} where $\mu>0$ controls the strength of the nonlinearity (and the fast–slow character for large $\mu$), and $$ f(x) = \frac{x^3}{3} - x. $$ {#eq-ode-2d-van-der-pol-2} ## Nullclines and Fixed Point The nullclines are especially simple in this model: $$ \dot x = 0 \quad \Longrightarrow \quad y = \frac{x^3}{3} - x, \qquad \dot y = 0 \quad \Longrightarrow \quad x = 0. $$ They intersect at the origin, so $(0,0)$ is the equilibrium point. For the parameter regime used in this session, the main long-run object of interest is not the equilibrium itself, but the attracting limit cycle around it. ```{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 amlab.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() ``` ## Implement the Vector Field ```python def van_der_pol(t, xy, mu=1.0): x, y = xy fx = x**3 / 3 - x dxdt = mu * (y - fx) dydt = -x / mu return np.array([dxdt, dydt]) ``` Follow the same pipeline as in [CDIMA Reaction](cdima.qmd): define the right-hand side, solve the initial value problem, compute nullclines, and then animate the resulting trajectory. ## 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 Role of $\mu$ The amplitude and period of the limit cycle depend on $\mu$. - For smaller $\mu$, the oscillation looks smoother and less sharply separated into time scales. - For larger $\mu$, the motion develops a clear slow-fast structure. - The trajectory spends long intervals drifting near the cubic nullcline, then jumps rapidly between branches. This is the characteristic relaxation-oscillation picture. ## What to Look For - Different initial conditions approach the same closed orbit. - The transient depends on the initial state, but the long-run behavior does not. - The time trace $x(t)$ reflects the same slow-fast pattern you see in the phase plane. Play with the parameter $\mu$ to see how it changes the shape of the limit cycle and the transient dynamics. ## What's Next? Once the geometry is clear, the next step is to animate the trajectory and, if needed, make the initial condition interactive. [Animation Template](animation.qmd){.btn .btn-primary} [Assignment](assignment.qmd){.btn .btn-secondary}