Lyapunov stable

A fixed point $x^{*}$ is Lyapunov stable if trajectories of nearby points $x$ remain close for future time. More formally the fixed point $x^{*}$ is Lyapunov stable, if for any $\epsilon>0$ , there is a $\delta>0$ such that for all $t\geq 0$ and for all $x\neq x^{*}$ it is verified

\displaystyle d(x^{*},x)<\delta\Rightarrow d(x^{*},x(t))<\epsilon.

In particular, $d(x^{*},x(0))=0$ .

Title	Lyapunov stable
Canonical name	LyapunovStable
Date of creation	2013-03-22 13:06:29
Last modified on	2013-03-22 13:06:29
Owner	armbrusterb (897)
Last modified by	armbrusterb (897)
Numerical id	10
Author	armbrusterb (897)
Entry type	Definition
Classification	msc 34D20
Synonym	Lyapunov stability
Synonym	Liapunov stable
Synonym	Liapunov stability
Related topic	AsymptoticallyStable
Related topic	AttractingFixedPoint
Related topic	StableFixedPoint
Related topic	NeutrallyStableFixedPoint
Related topic	UnstableFixedPoint