# 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 AsymptoticallyStable 2013-03-22 13:55:19 2013-03-22 13:55:19 Koro (127) Koro (127) 10 Koro (127) Definition msc 54H20 msc 37B99 UnstableFixedPoint LiapunovStable Lyapunov stable