Van der Pol Oscillator

Van der Pol Oscillator

The Van der Pol oscillator is a nonlinear system that models self-sustaining oscillations:

$x'' - \mu(1 - x^2)x' + x = 0$

• $\mu = 0$: reduces to simple harmonic motion
• $\mu > 0$: nonlinear damping

Nonlinear Damping

The term $-\mu(1 - x^2)x'$ is the key:

• When $|x| < 1$: the factor $(1 - x^2) > 0$, so damping is negative — the system adds energy. Small oscillations grow.
• When $|x| > 1$: the factor $(1 - x^2) < 0$, so damping is positive — the system removes energy. Large oscillations shrink.

This creates a limit cycle: regardless of starting conditions (except at rest), the system settles into a specific periodic orbit with amplitude $\approx 2$.

As a System

Let $v = x'$:

$\frac{dx}{dt} = v$ $\frac{dv}{dt} = \mu(1 - x^2)v - x$

Historical Context

The Van der Pol equation was originally developed to model vacuum tube circuits in early radio transmitters (1920s). It now models:

• Heartbeat rhythms (cardiac pacemakers)
• Circadian rhythms in biology
• Electrical oscillators
• Seizure dynamics in neuroscience

Your Task

Implement van_der_pol(mu, x0, v0, t_end, n) using forward Euler. Return (x, v).