limit cycle

Let

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

be a planar autonomous ordinary differential equation and $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ and the respective limit set is $\mathrm{\Gamma }$ then $\mathrm{\Gamma }$ is a limit cycle. In simpler terms a limit cycle is an isolated periodic solution of the system.
A limit cycle, $\mathrm{\Gamma }$, is a stable limit cycle (or $\omega$-limit cycle) if $\mathrm{\Gamma }$ is the $\omega$-limit set of all solutions in some neighborhood of $\mathrm{\Gamma }$.
A limit cycle, $\mathrm{\Gamma }$, is a unstable limit cycle (or $\alpha$-limit cycle) if $\mathrm{\Gamma }$ is the $\alpha$-limit set of all solutions in some neighborhood of $\mathrm{\Gamma }$.[PL]

Title	limit cycle
Canonical name	LimitCycle
Date of creation	2013-03-22 15:00:54
Last modified on	2013-03-22 15:00:54
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	9
Author	Daume (40)
Entry type	Definition
Classification	msc 34A12
Classification	msc 34C07
Synonym	$\omega$-limit cycle
Synonym	$\alpha$-limit cycle
Related topic	OmegaLimitSet
Defines	stable limit cycle
Defines	unstable limit cycle