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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 $\alpha$ -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]
- PL
- Perko, Lawrence: Differential Equations and Dynamical Systems (Third Edition). Springer, New York, 2001.
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"limit cycle" is owned by Daume.
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See Also: omega limit set
| Other names: |
 -limit cycle,  -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 13472 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) |
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Pending Errata and Addenda
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