Van der Pol Oscillator

The van der Pol oscillator is a nonlinear dynamical system described by a second-order differential equation with position-dependent damping. It was introduced by Balthasar van der Pol in 1927 to model the behavior of a triode circuit, and it has since become the canonical example of a system that produces relaxation oscillations when its nonlinearity parameter is large. For small damping, it behaves like a harmonic oscillator; for large damping, it exhibits the characteristic slow-fast dynamics that define the relaxation regime.

The equation is:

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

where \mu is the damping parameter. When \mu \gg 1, the system separates into slow and fast timescales, with the limit cycle hugging the cubic nullcline. This geometric structure is the prototype for all relaxation oscillations and the textbook application of Geometric Singular Perturbation Theory. The van der Pol oscillator also demonstrates that nonlinearity can stabilize: despite the damping term being negative for |x| < 1, the system does not blow up because the nonlinear term limits the amplitude.

The van der Pol equation belongs to the broader class of Liénard equations, which are planar systems with a single nonlinear restoring force and a position-dependent damping term.