The Van der Pol Oscillator: A Rhythm That Sets Its Own Amplitude

An oscillator that damps itself, and un-damps itself

A plain damped oscillator loses energy every cycle and its amplitude decays to zero. The Van der Pol oscillator, introduced by Dutch engineer Balthasar van der Pol in the 1920s while studying vacuum-tube circuits, does something stranger: its damping term has a sign that depends on amplitude. Small oscillations get pumped with energy; large oscillations get damped. The result is a system that settles onto one specific amplitude and stays there forever, regardless of where it started.

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

x     the state variable (e.g. circuit voltage)
mu    damping/nonlinearity parameter (mu >= 0)
x'    velocity, x''  acceleration

Look at the damping coefficient, -mu*(1 - x^2). When |x| < 1, the term (1 - x^2) is positive, but the overall sign is negative-of-negative -- energy is injected, and small oscillations grow. When |x| > 1, (1 - x^2) flips negative, damping turns positive, and large oscillations shrink. The crossover at |x| = 1 is a self-regulating amplitude control built directly into the equation.

The limit cycle

Whatever initial condition you start from -- a tiny nudge or a huge displacement -- the trajectory in phase space (x plotted against x') spirals onto the same closed loop. That closed loop is a limit cycle: an isolated periodic orbit that nearby trajectories are attracted to (or repelled from) rather than merely one member of a continuous family of orbits, the way a frictionless pendulum has infinitely many nested periodic orbits at every energy. The Van der Pol system has exactly one stable limit cycle, and the Poincare-Bendixson theorem guarantees that any bounded 2-D flow without a fixed point in its trapping region must approach a periodic orbit -- which is precisely what happens here, since the only fixed point, the origin, is unstable and repels trajectories outward into the cycle.

From sinusoid to relaxation oscillation

The parameter mu controls not just the size but the shape of that limit cycle. At mu = 0 the equation reduces to the plain harmonic oscillator x'' + x = 0, and technically the "limit cycle" degenerates into one of the harmonic oscillator's ordinary circular orbits -- a smooth sine wave. As mu grows past roughly 1, damping becomes strongly nonlinear and the waveform distorts into a slow, gentle rise punctuated by an almost instantaneous, sharp discharge -- a sawtooth-like relaxation oscillation. Physically this is a slow charge-up phase where the restoring force dominates, followed by a fast release once the trajectory crosses the threshold where damping flips sign. Neon-lamp relaxation oscillators and the heartbeat's own self-sustaining rhythm are classic real-world analogues of this two-timescale behaviour.

mu near 0     -> near-sinusoidal limit cycle, one timescale
mu around 1-3 -> visibly distorted, asymmetric oscillation
mu >> 1       -> relaxation oscillation: slow build, fast snap, two separated timescales

Why it matters beyond one vacuum tube

Van der Pol built the equation to explain the stable, self-sustaining oscillations of the triode circuits used in early radios -- circuits that, left alone, settle into one loud, clean amplitude no matter how they were switched on. But the same structure -- energy injection at small amplitude, dissipation at large amplitude -- shows up anywhere a system needs to sustain a rhythm autonomously: the FitzHugh-Nagumo model of a firing neuron is a close relative, cardiac pacemaker cells are modelled with Van der Pol-type equations, and it remains a standard textbook example in nonlinear dynamics precisely because it is the simplest system that exhibits a limit cycle you can compute by hand.

Integrating it numerically

Rewritten as a first-order system with y = x', the equation becomes x' = y, y' = mu*(1 - x^2)*y - x, ready for RK4 or any standard ODE stepper. The one subtlety is stiffness: for large mu the fast "snap" phase changes on a much shorter timescale than the slow buildup, so a fixed-step explicit integrator needs a small enough step to resolve the snap without blowing up, or an adaptive/implicit stepper if mu is pushed very high. The demo on this page uses a fixed small RK4 step, which is accurate and stable for the moderate mu range where the transition from sinusoidal to relaxation behaviour is most visually interesting.

Frequently asked questions

Try it live

Everything above runs in your browser — open Van der Pol Oscillator and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.