Uncertainty as to whether or not a system has a chaotic attractor

Abstract

Attracting chaotic behaviour in dynamical systems is often sensitive to small changes in parameters. If a perturbation in the parameter by a tiny amount ϵ can change the asymptotic behaviour of the system from being chaotic to being periodic, we call it parameter value ϵ-uncertain. Here, using a self-similar model of the intricate, intertwined parameter-space structure of the chaotic and periodic attractors, we investigate the scaling of this uncertainty with ϵ. We show that as ϵ approaches 0, the great majority of ϵ-uncertain parameters lie in high order windows, that is, windows within windows within windows …. The expected value of the order of the highest order window containing this parameter approaches infinity as ϵ goes to zero.