asymptotically stable

Let $(X,d)$ be a metric space and $f\colon X\to X$ a continuous function. A point $x\in X$ is said to be Lyapunov stable if for each $\epsilon>0$ there is $\delta>0$ such that for all $n\in\mathbb{N}$ and all $y\in X$ such that $d(x,y)<\delta$ , we have $d(f^{n}(x),f^{n}(y))<\epsilon$ .

We say that $x$ is asymptotically stable if it belongs to the interior of its stable set, i.e. if there is $\delta>0$ such that $\lim_{n\to\infty}d(f^{n}(x),f^{n}(y))=0$ whenever $d(x,y)<\delta$ .

In a similar way, if $\varphi\colon X\times\mathbb{R}\to X$ is a flow, a point $x\in X$ is said to be Lyapunov stable if for each $\epsilon>0$ there is $\delta>0$ such that, whenever $d(x,y)<\delta$ , we have $d(\varphi(x,t),\varphi(y,t))<\epsilon$ for each $t\geq 0$ ; and $x$ is called asymptotically stable if there is a neighborhood $U$ of $x$ such that $\lim_{t\to\infty}d(\varphi(x,t),\varphi(y,t))=0$ for each $y\in U$ .

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