Unstable dimension variability: A source of nonhyperbolicity in chaotic systems

E J Kostelich, I Kan, C Grebogi, E Ott, J A Yorke

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Abstract

The hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set. A familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable manifolds at points on the chaotic set. Here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible (respectively noninvertible) maps of dimension 3 (respectively 2) and higher. In particular, we investigate a situation (first considered by Abraham and Smale in 1970 for different purposes) in which the dimension of the unstable (and stable) tangent spaces are not constant over the chaotic set; we call this unstable dimension variability. A simple two-dimensional map that displays behavior typical of this phenomenon is presented and analyzed.

Original languageEnglish
Pages (from-to)81-90
Number of pages10
JournalPhysica. D, Nonlinear Phenomena
Volume109
Issue number1-2
Publication statusPublished - 1 Nov 1997

Keywords

  • hyperbolicity
  • stable manifolds
  • shadowing
  • DYNAMICAL-SYSTEMS
  • ATTRACTORS

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