Structural stability on manifolds with boundary

Clark Robinson*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

33 Scopus citations

Abstract

In application of dynamical systems, we are often interested in the motion in a compact subset of Euclidean space or possibly of a torus cross a Euclidean space. The natural way to study these problems is in a manifold with boundary, e.g., a disk. This paper gives sufficient conditions for a C1 flow ƒt on a compact manifold with boundary, M, to be weakly structurally stable. In particular, if ƒ has a hyperbolic chain-recurrent set (Axiom A) and satisfies the transversality condition then it is weakly structurally stable. Since we impose no assumptions on the type of tangencies of trajectories with the boundary of M, we need to weaken the notion of conjugacy to allow a homeomorphism h from M into a collared manifold M′ which contains M in its interior. Such a homeomorphism allows us to bypass the singularity theory used by Percell and Sotomayor. A similar result for diffeomorphisms of manifolds with boundary is obtained. If the flow satisfies a quadratic external boundary condition, then it is possible to demand that the conjugacy preserve the boundary. These results also allow an infinite chain-recurrent set rather than the finite set of periodic orbits allowed in the previous results on manifolds with boundary.

Original languageEnglish (US)
Pages (from-to)1-11
Number of pages11
JournalJournal of Differential Equations
Volume37
Issue number1
DOIs
StatePublished - 1980

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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