# Hartman-Grobman theorem

Consider the differential equation^{}

$${x}^{\prime}=f(x)$$ | (1) |

where $f$ is a ${C}^{1}$ vector field. Assume that ${x}_{0}$ is a hyperbolic equilibrium of $f$. Denote ${\mathrm{\Phi}}_{t}(x)$ the flow of (1) through $x$ at time $t$. Then there exists a homeomorphism $\phi (x)=x+h(x)$ with $h$ bouded, such that

$$\phi \circ {e}^{tDf({x}_{0})}={\mathrm{\Phi}}_{t}\circ \phi $$ |

is a sufficiently small neighboorhood of ${x}_{0}$.

This fundamental theorem in the qualitative analysis of nonlinear differential equations states that, in a small neighborhood^{} of ${x}_{0}$, the flow of the nonlinear equation (1) is qualitatively similar to that of the linear system ${x}^{\prime}=Df({x}_{0})x$.

Title | Hartman-Grobman theorem |
---|---|

Canonical name | HartmanGrobmanTheorem |

Date of creation | 2013-03-22 13:18:37 |

Last modified on | 2013-03-22 13:18:37 |

Owner | jarino (552) |

Last modified by | jarino (552) |

Numerical id | 4 |

Author | jarino (552) |

Entry type | Theorem |

Classification | msc 34C99 |