limit cycle


Let

x˙=f(x)

be a planar autonomousMathworldPlanetmath ordinary differential equationMathworldPlanetmath and Γ be a periodic solution of the system. If the α-limit set (http://planetmath.org/OmegaLimitSet) or the ω-limit set (http://planetmath.org/OmegaLimitSet) of a solution with initial value not on Γ and the respective limit set is Γ then Γ is a limit cycleMathworldPlanetmath. In simpler terms a limit cycle is an isolated periodic solution of the system.
A limit cycle, Γ, is a stable limit cycle (or ω-limit cycle) if Γ is the ω-limit set of all solutions in some neighborhood of Γ.
A limit cycle, Γ, is a unstable limit cycle (or α-limit cycle) if Γ is the α-limit set of all solutions in some neighborhood of Γ.[PL]

References

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 ω-limit cycle
Synonym α-limit cycle
Related topic OmegaLimitSet
Defines stable limit cycle
Defines unstable limit cycle