Unstable dimension variability: A source of nonhyperbolicity in chaotic systems

✍ Scribed by Eric J. Kostelich; Ittai Kan; Celso Grebogi; Edward Ott; James A. Yorke

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 706 KB
• Volume: 109
• Category: Article
• ISSN: 0167-2789
• DOI: 10.1016/s0167-2789(97)00161-9

✦ Synopsis

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.