asymptotically stable

Let (X,d) be a metric space and f:XX a continuous functionMathworldPlanetmath. A point xX is said to be Lyapunov stablePlanetmathPlanetmath if for each ϵ>0 there is δ>0 such that for all n and all yX such that d(x,y)<δ, we have d(fn(x),fn(y))<ϵ.

We say that x is asymptotically stable if it belongs to the interior of its stable set, i.e. if there is δ>0 such that limnd(fn(x),fn(y))=0 whenever d(x,y)<δ.

In a similar way, if φ:X×X is a flow, a point xX is said to be Lyapunov stable if for each ϵ>0 there is δ>0 such that, whenever d(x,y)<δ, we have d(φ(x,t),φ(y,t))<ϵ for each t0; and x is called asymptotically stable if there is a neighborhoodMathworldPlanetmathPlanetmath U of x such that limtd(φ(x,t),φ(y,t))=0 for each yU.

Title asymptotically stable
Canonical name AsymptoticallyStable
Date of creation 2013-03-22 13:55:19
Last modified on 2013-03-22 13:55:19
Owner Koro (127)
Last modified by Koro (127)
Numerical id 10
Author Koro (127)
Entry type Definition
Classification msc 54H20
Classification msc 37B99
Related topic UnstableFixedPoint
Related topic LiapunovStable
Defines Lyapunov stable