A New Approach To The Treatment Of Separatrix Chaos And Its Applications

TL;DR

This paper introduces a novel method to analyze separatrix chaos in weakly perturbed Hamiltonian systems, accurately describing the chaotic layer width and revealing new phenomena like facilitation of global chaos and expansion of stochastic webs.

Contribution

A new approach combining discrete and continuous dynamics to accurately describe separatrix chaos and predict novel phenomena in perturbed Hamiltonian systems.

Findings

Accurate description of maximum separatrix chaotic layer width.

Prediction of facilitation of global chaos onset.

Identification of large stochastic web growth.

Abstract

We consider time-periodically perturbed 1D Hamiltonian systems possessing one or more separatrices. If the perturbation is weak, then the separatrix chaos is most developed when the perturbation frequency lies in the logarithmically small or moderate ranges: this corresponds to the involvement of resonance dynamics into the separatrix chaos. We develop a method matching the discrete chaotic dynamics of the separatrix map and the continuous regular dynamics of the resonance Hamiltonian. The method has allowed us to solve the long-standing problem of an accurate description of the maximum of the separatrix chaotic layer width as a function of the perturbation frequency. It has also allowed us to predict and describe new phenomena including, in particular: (i) a drastic facilitation of the onset of global chaos between neighbouring separatrices, and (ii) a huge increase in the size of the low-dimensional stochastic web.