# stable manifold theorem

Let $E$ be an open subset of ${\mathbb{R}}^{n}$ containing the origin, let $f\in {C}^{1}(E)$, and let ${\varphi}_{t}$ be the flow of the nonlinear system ${x}^{\prime}=f(x)$.

Suppose that $f({x}_{0})=0$ and that $Df({x}_{0})$ has $k$ eigenvalues with negative real part and $n-k$ eigenvalues with positive real part. Then there exists a $k$-dimensional differentiable manifold $S$ tangent to the stable subspace ${E}^{S}$ of the linear system ${x}^{\prime}=Df(x)x$ at ${x}_{0}$ such that for all $t\ge 0$, ${\varphi}_{t}(S)\subset S$ and for all $y\in S$,

$$\underset{t\to \mathrm{\infty}}{lim}{\varphi}_{t}(y)={x}_{0}$$ |

and there exists an $n-k$ dimensional differentiable manifold $U$ tangent to the unstable subspace^{} ${E}^{U}$ of ${x}^{\prime}=Df(x)x$ at ${x}_{0}$ such that for all
$t\le 0$, ${\varphi}_{t}(U)\subset U$ and for all $y\in U$,

$$\underset{t\to -\mathrm{\infty}}{lim}{\varphi}_{t}(y)={x}_{0}.$$ |

Title | stable manifold theorem |
---|---|

Canonical name | StableManifoldTheorem |

Date of creation | 2013-03-22 12:57:17 |

Last modified on | 2013-03-22 12:57:17 |

Owner | jarino (552) |

Last modified by | jarino (552) |

Numerical id | 4 |

Author | jarino (552) |

Entry type | Theorem |

Classification | msc 34C99 |