## Abstract

In this paper, we show that the qualitative property of a Morse Smale gradient-like flow is preserved by its discretization mapping obtained via numerical methods. This means that for all sufficiently small h. there is a homeomorphisrn H_{h} conjugating the time-h map Φ^{h} of the flow to the discretization mapping φ^{h}. Garay [Numer. Math., 72 (1996), pp. 449-479] showed this result by relying on techniques of Robbin [Ann. Math., 94 (1971), pp. 447-493]. Our result sharpens and unifies that in [Numer. Math., 72 (1996), pp. 449-479] by using Robinson's method in [J. Differential Equations, 22 (1976), pp. 28-73] of the structural stability theorem for diffeomorphisrns. We also study the problem on a manifold with boundary. Under the assumption that the manifold M is positively invariant for the flow, we show that the qualitative properties are weakly stable, which means we allow the homeomorphism H_{h} from M into a larger manifold M′ which contains M and is of the same dimension as M.

Original language | English |
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Pages (from-to) | 381-388 |

Number of pages | 8 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 1997 |

## Keywords

- Gradient-like flow
- Morse-Smale flow
- Numerical method
- Structural stability