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

Abstract

This paper examines the most probable route to chaos 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.