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.