Limit Cycle
An isolated closed orbit that a nonlinear system generates and maintains entirely on its own — the mathematics of self-sustained rhythm.
Limit Cycle
A limit cycle is an isolated closed loop in state space that nearby trajectories spiral onto. Unlike the Simple Harmonic Oscillator's family of nested orbits — where the amplitude is set by how hard you started it — a limit cycle has one preferred amplitude that the system returns to no matter where it begins. It is an Attractor, but a one-dimensional, oscillating one.
This is the mathematics of self-sustained oscillation: a heartbeat, a firing neuron, a chirping circuit, a beating laser. None of these needs a periodic external drive; the rhythm is intrinsic, born from a balance between an internal energy source and dissipation.
The van der Pol oscillator
The classic example is the van der Pol oscillator, originally a model of a vacuum-tube circuit:
The trick is in the damping term -\mu(1-x^2)\dot x. When the amplitude is small (x^2 < 1) the term is negative damping — it pumps energy in and the oscillation grows. When the amplitude is large (x^2 > 1) it becomes ordinary positive damping, bleeding energy away. The system is squeezed from both sides onto a single stable loop.
Why limit cycles need nonlinearity
A linear system can only spiral in (damped) or out (unstable) — it can never settle onto a fixed-amplitude loop. Limit cycles are inescapably nonlinear: it takes a term like \mu(1-x^2) that changes sign with amplitude to trap the motion. They typically appear when a Fixed Point loses Stability through a Hopf Bifurcation, spinning a stable equilibrium off into a growing oscillation as a parameter crosses a threshold.