be an autonomousMathworldPlanetmath ordinary differential equationMathworldPlanetmath defined by the vector field f:VV then x(t)V a solution of the system is a cycle(or periodic solution) if it is a closed solution which is not an equilibrium point. The period of a cycle is the smallest positive T such that x(t)=x(t+T).
Let ϕt(x) be the flow defined by the above ODE and d be the metric of V then:
A cycle, Γ, is a stable cycle if for all ϵ>0 there exists a neighborhood U of Γ such that for all xU, d(ϕt(x),Γ)<ϵ.
A cycle, Γ, is unstable cycle if it is not a stable cycle.
A cycle, Γ, is asymptotically stable cycle if for all xU where U is a neighborhood of Γ, limtd(ϕt(x),Γ)=0.[PL]


x˙ = -y
y˙ = x

then the above autonomous ordinary differential equations with initial value condition (x(0),y(0))=(1,0) has a solution which is a stable cycle. Namely the solution defined by

x(t) = cost
y(t) = sint

which has a period of 2π.


Title cycle
Canonical name Cycle12
Date of creation 2013-03-22 15:00:51
Last modified on 2013-03-22 15:00:51
Owner Daume (40)
Last modified by Daume (40)
Numerical id 6
Author Daume (40)
Entry type Definition
Classification msc 34A12
Classification msc 34C07
Synonym periodic solution
Synonym stable periodic solution
Synonym unstable periodic solution
Synonym asymptotically stable periodic solution
Defines period
Defines stable cycle
Defines unstable cycle
Defines asymptotically stable cycle