# Limit cycle

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A limit cycle in a dynamical system is a closed orbit in phase space, which is the limiting behavior of some trajectories either as $t\rightarrow\infty$ or $t\rightarrow-\infty$. 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 $t\rightarrow\infty$. Conversely, trajectories in the immediate neighborhood of an unstable limit cycle only approach the cycle in the limit $t\rightarrow-\infty$, and thus diverge from it in forward time.