limit cycleLimitCycle2005-02-06 21:34:362007-01-02 03:53:27Definitionstable limit cycleunstable limit cycle% this is the default PlanetMath preamble. as your knowledge
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Let
$$\dot{x}=f(x)$$
be a planar autonomous ordinary differential equation and $\Gamma$ be a periodic solution of the system. If the \PMlinkname{$\alpha$-limit set}{OmegaLimitSet} or the \PMlinkname{$\omega$-limit set}{OmegaLimitSet} of a solution with initial value not on $\Gamma$ and the respective limit set is $\Gamma$ then $\Gamma$ is a \emph{limit cycle}. In simpler terms a limit cycle is an isolated periodic solution of the system.\\
A limit cycle, $\Gamma$, is a \emph{stable limit cycle} \textit{(or \emph{$\omega$-limit cycle})} if $\Gamma$ is the $\omega$-limit set of all solutions in some neighborhood of $\Gamma$.\\
A limit cycle, $\Gamma$, is a \emph{unstable limit cycle} \textit{(or \emph{$\alpha$-limit cycle})} if $\Gamma$ is the $\alpha$-limit set of all solutions in some neighborhood of $\Gamma$.\cite{PL}
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\bibitem[PL]{PL} Perko, Lawrence: Differential Equations and Dynamical Systems \textit{(Third Edition)}. Springer, New York, 2001.
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