PlanetMath: limit cycle

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limit cycle

(Definition)

Let

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

be a planar autonomous ordinary differential equation and $\Gamma$ be a periodic solution of the system. If the $\alpha$ -limit set or the $\omega$ -limit set 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

PL: Perko, Lawrence: Differential Equations and Dynamical Systems (Third Edition). Springer, New York, 2001.

"limit cycle" is owned by Daume.

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See Also: omega limit set

Other names:

$\omega$ -limit cycle, $\alpha$ -limit cycle

Also defines:

stable limit cycle, unstable limit cycle

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Cross-references: neighborhood, isolated, limit, solution, periodic solution, ordinary differential equation, autonomous, planar
There are 7 references to this entry.

This is version 6 of limit cycle, born on 2005-02-06, modified 2007-01-02.
Object id is 6722, canonical name is LimitCycle.
Accessed 9467 times total.

Classification:

AMS MSC:	34C07 (Ordinary differential equations :: Qualitative theory :: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramif)
	34A12 (Ordinary differential equations :: General theory :: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions)

Pending Errata and Addenda

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