# Limit cycle

(under construction)

Original entry: Ian Burgess, Fall 2009

## Definition

A limit cycle in a dynamical system is a closed orbit in phase space, which is the limiting behavior of some trajectories either as <math>t\rightarrow\infty</math> or <math>t\rightarrow-\infty</math>. Stability can be attributed to a limit cycle just as with an equilibrium point. A limit cycle is stable if all trajectories in the immediate neighborhood of the limit cycle return to the cycle at <math>t\rightarrow\infty</math>. Conversely, trajectories in the immediate neighborhood of an unstable limit cycle only approach the cycle in the limit <math>t\rightarrow-\infty</math>, and thus diverge from it in forward time.

## References

1. S.H. Strogatz, *Nonlinear Dynamics and Chaos* Westview Press 2000.