Structural stability and hyperbolicity violation in high-dimensional dynamical systems

revised version: June 2006
David Albers, and J. Sprott
(Please use for correspondence this email).

Submission date: 24. Oct. 2005
Pages: 40
published in: Nonlinearity, 19 (2006) 8, p. 1801-1847 
DOI number (of the published article): 10.1088/0951-7715/19/8/005
MSC-Numbers: 37XX
PACS-Numbers: 05.45.-a, 89.75.fb, 89.75.-k
Keywords and phrases: structural stability, partial hyperbolicity, stability conjecture, high-dimensional dynamical systems
Download preprint: PDF (2103 kB), PS ziped (1657 kB)

Abstract:
This report investigates the dynamical stability conjectures of Palis and Smale, and Pugh and Shub from the standpoint of numerical observation and lays the foundation for a stability conjecture. As the dimension of a dissipative dynamical system is increased, it is observed that the number of positive Lyapunov exponents increases monotonically, the Lyapunov exponents tend towards continuous change with respect to parameter variation, the number of observable periodic windows decreases (at least below numerical precision), and a subset of parameter space exists such that topological change is very common with small parameter perturbation. However, this seemingly inevitable topological variation is never catastrophic (the dynamic type is preserved) if the dimension of the system is high enough.