Van der Pol oscillator
A nonlinear oscillator with amplitude-dependent damping. Near the origin, energy is pumped in; far from it, energy is dissipated. The result is a limit cycle — every trajectory, regardless of initial conditions, converges to the same closed orbit. As μ increases, the smooth sinusoid sharpens into a relaxation oscillation.
ẍ − μ(1 − x²)ẋ + x = 0 • Equivalent: dx/dt = y, dy/dt = μ(1 − x²)y − x
The Van der Pol oscillator
Balthasar van der Pol studied this equation in the 1920s while working on vacuum tube circuits at Philips. The key feature is nonlinear damping: when |x| < 1, the damping coefficient is negative (energy is added), and when |x| > 1, damping is positive (energy is removed). This creates a self-sustaining oscillation at a specific amplitude regardless of initial conditions.
The limit cycle
In the phase plane (x vs dx/dt), all trajectories spiral toward the same closed curve — the limit cycle. Points starting inside the cycle spiral outward; points starting outside spiral inward. This is an attractor: the asymptotic behavior is independent of initial conditions (except the unstable fixed point at the origin).
The μ parameter
When μ is small, the system is nearly linear and oscillates sinusoidally. As μ increases, the oscillation becomes increasingly nonlinear — the system spends long periods near x ≈ ±2, then rapidly transitions between them. These are relaxation oscillations, named for the slow “relaxation” phases between fast jumps. At large μ, the period grows approximately as (3 − 2 ln 2)μ.
Applications
The Van der Pol oscillator appears in models of heartbeat rhythms, circadian cycles, neural firing, and electronic oscillator circuits. It was one of the first systems used to study self-excited oscillations and helped establish the field of nonlinear dynamics.