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]
References
- PL Perko, Lawrence: Differential Equations and Dynamical Systems^{} (Third Edition). Springer, New York, 2001.
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 |