Routes to chaos in high-dimensional dynamical systems: a qualitative numerical study

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

Submission date: 24. Oct. 2005
Pages: 32
published in: Physica / D, 223 (2006) 2, p. 194-207 
DOI number (of the published article): 10.1016/j.physd.2006.09.004
MSC-Numbers: 37XX
PACS-Numbers: 05.45.-a, 89.75.-k
Keywords and phrases: routes to chaos, bifurcation theory, high-dimensional dynamics
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Abstract:
This paper examines the most probable route to chaos in high-dimensional dynamical systems in a general computational setting (time-delay neural networks). The most probable route to chaos in high-dimensional, discrete-time maps (relative to our construction) is observed to be a sequence of Neimark-Sacker bifurcations into chaos. A means for determining and understanding the degree to which the Landau-Hopf route to turbulence is non-generic in the space of mappings is outlined. Finally, a scenario regarding the onset of chaos in high-dimensional dissipative dynamical systems where the strongly stable directions with rotation decouple before chaos onsets is presented.