Van der Pol Oscillator

What you'll learn

Nonlinear dynamics and the concept of a limit cycle
How a single parameter can change the character of oscillations
Why initial conditions don't matter for the steady-state behavior

The Van der Pol oscillator is a classic nonlinear system. Unlike the mass-spring-damper, it has a limit cycle — regardless of where you start, the system settles into the same periodic orbit. The parameter μ controls the strength of the nonlinearity: small values give near-sinusoidal oscillations, while large values produce sharp relaxation oscillations with long plateaus and fast transitions.

\frac{d x}{d t} \frac{d y}{d t} = y = μ (1 - x^{2}) y - x

The nonlinear damping term $m u (1 - x^{2}), y$ is the key: when $∣ x ∣ < 1$ it adds energy (negative damping), and when $∣ x ∣ > 1$ it removes energy (positive damping). This push-pull is what creates the stable limit cycle.

Things to try

Set mu = 0.1 — the oscillation looks nearly sinusoidal.
Crank it up to mu = 5.0 — notice the sharp sawtooth shape of relaxation oscillations.
Change the plot's X axis from time to x and plot y against it — this phase portrait reveals the limit cycle as a closed loop, with the trajectory spiraling onto it from wherever it starts.
Change the initial conditions to x(start = 0.01) and y(start = 0.01) — the system still converges to the same limit cycle, just takes longer to get there.
Try x(start = 5) to start outside the limit cycle — the amplitude shrinks instead of growing.

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