Tool

Van der Pol oscillator

Self-sustained nonlinear oscillator with negative damping for small x. The unforced version has a unique stable limit cycle (Liénard); the forced version exhibits mode-locking tongues and chaos.

ẍ − μ (1 − x²) ẋ + x = A cos(ω t)

unforced (A = 0): unique stable limit cycle (Liénard's theorem).
forced (A > 0): mode-locking tongues + chaos at large μ.
3rd axis is the drive phase φ = ω t (wrapped to [−π, π]).

Background at /flows/vanderpol.