Limit Cycle

An isolated closed periodic trajectory in a nonlinear system that nearby states get pulled toward and circle around.

A limit cycle is a concept from nonlinear dynamics, with the systematic study of it starting with Poincaré: in a state space (such as an angle-versus-angular-velocity plane), it's a closed periodic trajectory that nearby trajectories spiral toward over time (a stable limit cycle) or spiral away from (an unstable one). A stable limit cycle means the system spontaneously settles into an oscillation at a fixed rhythm, and returns to that same rhythm even after a disturbance; the classic example is the Van der Pol oscillator. Walking-robot research treats a stable periodic gait as a stable limit cycle: McGeer's passive-dynamic walking robots have no motors at all, and walk stably down a shallow slope powered only by gravity. Analysis typically takes the state at each foot-touchdown instant to build a Poincaré map (a return map), turning the question of whether a gait is stable into whether this discrete map has a stable fixed point.

ExampleAn unpowered 'rimless wheel' (a wheel with spokes but no rim) placed on a slope will, across a fairly wide range of starting speeds, quickly converge to the same stable rolling rhythm — a stable limit cycle, and a classic example from Tedrake's Underactuated Robotics.

Also called

Stable Periodic Orbit, Attracting Periodic Orbit

Related

Passive Dynamic Walking · Gait · Central Pattern Generator · Hybrid Zero Dynamics · Lyapunov Stability · Gait Phase (Phase Clock)

Sources

Wikipedia: Limit cycle
Tedrake, Underactuated Robotics: Simple Models of Walking and Running