limit cycle

A heartbeat, the firing of a neuron, the steady buzz of a violin string driven by a bow: these are oscillations that settle onto a definite rhythm and amplitude no matter how they start, and that return to that same rhythm after a disturbance. A limit cycle is the phase-space object behind such self-correcting, self-sustained oscillation, an isolated closed loop that nearby trajectories spiral toward (or away from).

A limit cycle is a closed (periodic) trajectory in phase space that is isolated, meaning no other closed trajectory lies arbitrarily close to it. This isolation is what distinguishes it from the closed orbits of a conservative system like the frictionless pendulum, which come in a continuous nested family and whose amplitude is set entirely by the initial conditions. A stable (attracting) limit cycle pulls in all nearby trajectories, so the oscillation's period and amplitude are intrinsic to the system, set by its equations rather than by how it was launched. Limit cycles require nonlinearity; they are impossible in linear systems and, by a theorem, in one-dimensional flows.

The archetype is the van der Pol oscillator, d^2x/dt^2 - mu (1 - x^2) dx/dt + x = 0, in which a term that pumps energy in at small amplitude and drains it at large amplitude balances to give one preferred cycle. Where you meet them: relaxation oscillators, circadian clocks, the Belousov-Zhabotinsky reaction, and the onset of many instabilities. A limit cycle is often born when a fixed point loses stability in a Hopf bifurcation. Do not confuse a limit cycle with the neutrally stable orbits of a conservative system; those are not isolated and do not restore their amplitude after a kick.