Van der Pol Oscillator

The Van der Pol oscillator is a nonlinear second order differential equation describing an oscillator with damping that varies with amplitude, injecting energy into small oscillations while removing energy from large ones, so that the system settles into a single, stable, self-sustaining oscillation known as a limit cycle regardless of its exact starting point. It is named after the Dutch electrical engineer Balthasar van der Pol, who introduced the equation in the 1920s while studying the behavior of electrical circuits built from vacuum tubes. In experiments with a forced version of his circuit in 1927, van der Pol and his collaborator Jan van der Mark observed irregular, noise-like oscillations that were not understood at the time but are now recognized as an early experimental glimpse of deterministic chaos, decades before chaos theory was formally developed, making the Van der Pol oscillator one of the earliest physical systems in which such behavior was recorded.