Structural stability of Morse-Smale gradient-like flows under discretizations

Ming-Chia Li*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

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 Hh conjugating the time-h map Φh of the flow to the discretization mapping φh. 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 Hh from M into a larger manifold M′ which contains M and is of the same dimension as M.

Original languageEnglish
Pages (from-to)381-388
Number of pages8
JournalSIAM Journal on Mathematical Analysis
Volume28
Issue number2
DOIs
StatePublished - 1 Jan 1997

Keywords

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

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