Van der Pol Oscillator

The Van der Pol Equation

The Van der Pol oscillator, introduced by Dutch physicist Balthasar van der Pol (1920), models self-sustained oscillations in electrical circuits with nonlinear damping. The second-order ODE is:

Writing \(v = \dot{x}\), this is equivalent to the first-order system \(\dot{x}=v\), \(\dot{v}=\mu(1-x^2)v - x\).

Physical Interpretation — Horizontal Slider

The animation shows a mass block threaded onto a fixed horizontal pole. The restoring force \(-x\) acts like a spring pulling the mass back to center. The nonlinear damping \(\mu(1-x^2)\dot{x}\) switches sign depending on position:

• Blue zone \(|x|<1\): negative damping — the system pumps energy in, like a hidden motor accelerating the mass.
• Red zone \(|x|>1\): positive damping — energy is removed, braking the mass.

These two competing effects self-regulate to drive every trajectory toward the unique stable limit cycle of amplitude ≈ 2.

Regimes

• \(\mu = 0\): Simple harmonic oscillator — sinusoidal motion, constant amplitude.
• \(\mu \ll 1\): Quasi-sinusoidal, slowly converging to limit cycle. Period \(\approx 2\pi\).
• \(\mu \approx 1\): Clear nonlinear distortion, rapid approach to limit cycle.
• \(\mu \gg 1\): Relaxation oscillations — the mass drifts slowly then snaps rapidly. Period \(T \approx 1.614\,\mu\).

Numerical Solvers

• RK4 (classical, order 4) — used for \(\mu \le 5\). Four evaluations per step. Fixed step \(\Delta t = 0.01/(1+\mu/5)\).
• SDIRK2 (singly diagonally implicit, order 2) — used for \(\mu > 5\). A-stable and L-stable: unconditionally stable for any step size. Newton iteration with analytic Jacobian and exact \(2\times2\) solve. Adaptive step control.