The Van der Pol oscillator is the nonlinear system

where controls the strength of the nonlinear damping.

The only equilibrium is at . Linearizing near the origin gives

so the eigenvalues satisfy . In particular, when we have , and as crosses the real part of the eigenvalues changes sign. This change in stability is the local signature of a Hopf bifurcation.

For the origin is a stable spiral and trajectories decay to . For the origin becomes unstable, but solutions do not blow up: the term injects energy at small amplitudes and dissipates energy at large amplitudes, producing a stable limit cycle.

In this demo, the trajectory is computed in Rust, compiled to WebAssembly, and rendered in your browser using a <canvas>.

Try refreshing and watch how trajectories spiral away from the origin and settle onto the same closed orbit. Next we’ll add a slider for (and optionally initial conditions) to make the Hopf transition visible directly in the phase portrait.