# 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 $\u03f5>0$ 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 $\phi :X\times \mathbb{R}\to X$ is a flow, a point $x\in X$ is said to be Lyapunov stable if for each $\u03f5>0$ 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 $y\in U$.

Title | asymptotically stable |
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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 |