mathematics//dynamical systems//equilibrium and stability//limit cycle

A limit cycle is an oscillation a system settles into by itself, with an amplitude and a period fixed by the system rather than by where it started: a heart's beat, a clock's escapement, a violin string under the bow, and, unwanted, a control valve hunting back and forth around its setpoint. In the phase portrait it is an isolated closed orbit, and nearby trajectories spiral onto it (a stable limit cycle) or away from it (an unstable one).

A limit cycle is an oscillation a system settles into by itself, with an amplitude and a period fixed by the system rather than by where it started: a heart's beat, a clock's escapement, a violin string under the bow, and, unwanted, a control valve hunting back and forth around its setpoint. In the phase portrait it is an isolated closed orbit, and nearby trajectories spiral onto it (a stable limit cycle) or away from it (an unstable one).

It needs nonlinearity and a source of energy. A linear oscillator can also run round a closed orbit, but every amplitude is possible and a nudge changes the amplitude for good; a limit cycle pulls the amplitude back after a nudge, because the system gains energy when the swing is small and loses it when the swing is large. The Van der Pol oscillator, first written for vacuum-tube circuits, is the textbook case:

x¨−μ (1−x2) x˙+x=0,μ>0.\ddot x-\mu\,(1-x^2)\,\dot x+x=0,\qquad \mu>0.x¨−μ(1−x2)x˙+x=0,μ>0.

Its damping is negative for ∣x∣<1|x|<1∣x∣<1, which pumps small swings up, and positive for ∣x∣>1|x|>1∣x∣>1, which drains large ones, so every trajectory except the equilibrium ends on the same cycle.

In control loops a limit cycle is usually a fault. A valve with stiction sticks until the controller's integral has pushed hard enough, then jumps past the target, and the loop cycles for ever at a constant amplitude; a saturated actuator in a loop tuned too hot can do the same. A steady oscillation in a plant's trend, with a triangular controller output and a measurement shaped like a square wave, is the classic signature of a sticking valve.

On-off control makes one by design. A thermostat with a dead band cycles the room temperature between two values, with a period and amplitude set by the dead band and the room's lag (controller). Relay autotuning of PID controllers uses exactly this oscillation to measure the plant (PID tuning).

Only a cycle that keeps its amplitude is a limit cycle. An oscillation that grows without bound is instability, one that dies out is a stable spiral, and telling the three apart on a short recording is not always possible.