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 language | English |
---|---|
Pages (from-to) | 81-90 |
Number of pages | 10 |
Journal | Physica. D, Nonlinear Phenomena |
Volume | 109 |
Issue number | 1-2 |
Publication status | Published - 1 Nov 1997 |
Keywords
- hyperbolicity
- stable manifolds
- shadowing
- DYNAMICAL-SYSTEMS
- ATTRACTORS