the van der Pol oscillator
/ van der POLL /
An ordinary oscillator either rings down to silence (with friction) or, in the idealized frictionless case, repeats with whatever amplitude you gave it. The van der Pol oscillator does something neither of those: whatever amplitude you start with — big or tiny — it converges to one and the same steady oscillation, chosen entirely by the system itself. It is the textbook model of a self-sustained rhythm, born from studying vacuum-tube electronic circuits.
The equation is x'' - mu(1 - x^2) x' + x = 0, with a positive parameter mu controlling the strength of the effect. Look at the middle term as a strange, state-dependent damping with coefficient -mu(1 - x^2). When x is small (|x| < 1) the factor (1 - x^2) is positive, so the damping is negative — it pumps energy in and small oscillations grow. When x is large (|x| > 1) the factor is negative, so the damping is positive — it bleeds energy out and large oscillations shrink. The system is forever feeding small swings and starving large ones, and it balances in between, on a single stable limit cycle. Poincare-Bendixson can be used to prove that cycle exists; every trajectory but the unstable origin spirals onto it.
Through this field's lens, van der Pol is the showcase nonlinear example: it has an isolated, attracting periodic orbit that no linear system could ever produce, and its existence is established by trapping-region arguments and the Poincare-Bendixson theorem rather than by any formula. For small mu the cycle is nearly a smooth circle; for large mu it becomes a sharp 'relaxation oscillation' — long slow build-ups punctuated by sudden jumps — which models the firing of neurons, the beating of the heart, and many electronic and biological clocks.
For x'' - (1 - x^2) x' + x = 0 (mu = 1), both a trajectory starting near the origin (growing outward) and one starting far out (shrinking inward) spiral onto the same closed loop — the unique stable limit cycle.
Negative damping for small x and positive damping for large x squeeze every start onto one self-chosen oscillation.
The amplitude is set by the system, not by the initial conditions — a defining feature of a limit cycle and impossible in any linear constant-coefficient equation. Note the origin is an unstable equilibrium, the lone trajectory that does not reach the cycle.