Structural stability for the Euler method

Ming-Chia Li*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


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 languageEnglish
Pages (from-to)747-755
Number of pages9
JournalSIAM Journal on Mathematical Analysis
Issue number4
StatePublished - 1 Jan 1999


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

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