asymptotically stable

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

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 ${lim}_{n\to \mathrm{\infty }}d(f^n(x),f^n(y))=0$ whenever .

In a similar way, if $\phi :X\times ℝ\to X$ is a flow, a point $x\in X$ is said to be Lyapunov stable if for each $ϵ>0$ there is $\delta >0$ such that, whenever , we have for each $t\ge 0$; and $x$ is called asymptotically stable if there is a neighborhood $U$ of $x$ such that ${lim}_{t\to \mathrm{\infty }}d(\phi (x,t),\phi (y,t))=0$ for each $y\in U$.