Structural Stability and Hyperbolicity Violation in High-Dimensional Dynamical Systems

D. J. Albers and J. C. Sprott
Physics Department, University of Wisconsin, Madison, WI 53706, USA

Received 24 October 2005, in final form 6 June 2006
Published 7 July 2006


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 in high enough.

Ref: D. J. Albers and J. C. Sprott, Nonlinearity 19, 1801-1847 (2006)

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