limit cycle

Let

 $\dot{x}=f(x)$

be a planar autonomous ordinary differential equation and $\Gamma$ be a periodic solution of the system. If the $\alpha$-limit set (http://planetmath.org/OmegaLimitSet) or the $\omega$-limit set (http://planetmath.org/OmegaLimitSet) of a solution with initial value not on $\Gamma$ and the respective limit set is $\Gamma$ then $\Gamma$ is a limit cycle. In simpler terms a limit cycle is an isolated periodic solution of the system.
A limit cycle, $\Gamma$, is a stable limit cycle (or $\omega$-limit cycle) if $\Gamma$ is the $\omega$-limit set of all solutions in some neighborhood of $\Gamma$.
A limit cycle, $\Gamma$, is a unstable limit cycle (or $\alpha$-limit cycle) if $\Gamma$ is the $\alpha$-limit set of all solutions in some neighborhood of $\Gamma$.[PL]

References

Title limit cycle LimitCycle 2013-03-22 15:00:54 2013-03-22 15:00:54 Daume (40) Daume (40) 9 Daume (40) Definition msc 34A12 msc 34C07 $\omega$-limit cycle $\alpha$-limit cycle OmegaLimitSet stable limit cycle unstable limit cycle