Limit Cycle

In mathematics, in the area of dynamical systems, a limit cycle on a plane or a two-dimensional manifold is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. In the case where all the neighbouring trajectories approach the limit cycle as time approaches infinity, it is called a stable or attractive limit cycle (ω-limit cycle). If instead all neighbouring trajectories approach it as time approaches negative infinity, it is an unstable or non-attractive limit cycle (α-limit cycle).

Stable limit cycles imply self-sustained oscillations. Any small perturbation from the closed trajectory would cause the system to return to the limit cycle, making the system stick to the limit cycle.

As seen in the figure, trajectories for various initial states of this system converge to the limit cycle. Hence, this system exhibits self-sustained oscillations.

The number of limit cycles of a polynomial differential equation is the main object of the second part of Hilbert's sixteenth problem. Bendixson's theorem and the Poincaré–Bendixson theorem predict the absence or existence, respectively, of limit cycles of two-dimensional nonlinear dynamical systems.