# omega limit set

Let $\mathrm{\Phi}(t,x)$ be the flow of the differential equation^{} ${x}^{\prime}=f(x)$, where $f\in {C}^{k}(M,{\mathbb{R}}^{n})$, with $k\ge 1$ and $M$ an open subset of ${\mathbb{R}}^{n}$.
Consider $x\in M$.

The omega limit set of $x$, denoted $\omega (x)$, is the set of points $y\in M$ such that there exists a sequence ${t}_{n}\to \mathrm{\infty}$ with $\mathrm{\Phi}({t}_{n},x)=y$.

Similarly, the alpha limit set of $x$, denoted $\alpha (x)$, is the set of points $y\in M$ such that there exists a sequence ${t}_{n}\to -\mathrm{\infty}$ with $\mathrm{\Phi}({t}_{n},x)=y$.

Note that the definition is the same for more general dynamical systems^{}.

Title | omega limit set |
---|---|

Canonical name | OmegaLimitSet |

Date of creation | 2013-03-22 13:18:42 |

Last modified on | 2013-03-22 13:18:42 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 37B99 |

Classification | msc 34C05 |

Synonym | $\omega $-limit set |

Synonym | $\alpha $-limit set |

Related topic | LimitCycle |

Defines | alpha limit set |