## Abstract

In this paper, we show that if a flow φ
^{t}
has a hyperbolic chain recurrent set either without fixed points or with only fixed points, and satisfies the strong transversality condition, then φ
^{t}
is structurally stable with respect to numerical methods, including the Euler method, which was not done in [B. M. Garay, Numer. Math., 72 (1996), pp. 449-479], [M.-C. Li, J. Differential Equations, 141 (1997), pp. 1-12], and [M.-C. Li, SIAM J. Math. Anal., 28 (1997), pp. 381-388]. The proof is an application of the invariant manifold techniques developed by Hirsch, Pugh, and Shub [M. Hirsh, C. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer-Verlag, New York, 1977] and Robinson [C. Robinson, J. Differential Equations. 22 (1976), pp. 28-73]. The result is an extension of our previous work [M.-C. Li, Proc. Amer. Math. Soc., 127 (1999), pp. 289-295].

Original language | English |
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Pages (from-to) | 747-755 |

Number of pages | 9 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 30 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 1999 |

## Keywords

- Chain recurrent set
- Euler method
- Hyperbolicity
- Numerical method
- Structural stability