# Lyapunov stable

A fixed point $x^{*}$ is 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