Van der Pol: Limit Cycle to Relaxation Oscillator

What you are seeing: the Van der Pol equation $\ddot{x} - \mu (1 - x^2) \dot{x} + x = 0$ . For any starting point off the trivial fixed point at the origin, the trajectory converges to a unique limit cycle. At small $\mu$ the cycle is nearly circular (close to the harmonic oscillator). As $\mu$ grows, the cycle deforms into a relaxation oscillation: long slow phases interrupted by fast jumps.

Left panel: the phase portrait $(x, v)$ with the trajectory tracing the limit cycle. Right panel: the time series $x(t)$ showing the characteristic slow-fast pattern when $\mu$ is large. Slide $\mu$ from 0 (pure SHO) to 8 (deep relaxation) and watch the cycle deform.

Figure 1. Van der Pol oscillator. Method: RK4 integration with phase-portrait and time-series rendering.

mu1.00

speed3

WHAT TO TRY

Raise mu: the Van der Pol limit cycle morphs from a near-circular orbit (mu small, almost a harmonic oscillator) into a sharp relaxation oscillation with fast jumps and slow crawls. The x(t) trace squares off.
Start the trajectory anywhere off the origin: it always spirals onto the same limit cycle, the self-sustained oscillation that does not depend on the initial condition.
Watch the phase portrait at large mu: the orbit hugs the cubic nullcline then snaps across, the stick-slip rhythm behind heartbeats, neuron firing and stick-slip friction.