Routes to Chaos in High-Dimensional Dynamical Systems: A Qualitative Numerical Study

D. J. Albers and J. C. Sprott

Max Planck Institute for Mathematics in the Sciences, D-04103 Leipzig, Germany
Department of Physics, University of Wisconsin, Madison, 1150 University Avenue, Madison, WI 53706-1390, United States
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, United States
Computational Science and Engineering Center, University of California, Davis, One Shields Ave, Davis, CA 95616, United States

Received 31 July 2004; received in revised form 28 April 2006; accepted 7 September 2006

ABSTRACT

This paper examines the most probable route to chaos [in] a high-dimensional dynamical systems function space (time-delay neural networks) endowed with a probability measure in a computational setting. The most probable route to chaos (relative to the measure we impose on the function space) as the dimension is increased is observed to be a sequence of Neimark-Sacker bifurcations into chaos. The analysis is composed of the study of an example dynamical system followed by a probabilistic study of the ensemble of dynamical systems from which the example was drawn. A scenario depicting the decoupling of the stable manifolds of the torus leading up to the onset of chaos in high-dimensional dissipative dynamical systems is also presented.

Ref: D. J. Albers and J. C. Sprott, Physica D 223, 194-207 (2006)

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