Disentangling regular and chaotic motion in the standard map using complex network analysis of recurrences in phase space

TL;DR

This paper uses recurrence network analysis of phase space to distinguish between regular and chaotic orbits in the standard map, providing a new geometric approach to chaos detection in conservative systems.

Contribution

It introduces recurrence network analysis as a novel method to differentiate regular and chaotic regimes in the standard map, especially in challenging laminar phases.

Findings

Recurrence network properties differ between regular and chaotic orbits.

Chaotic orbits during laminar phases have distinct geometric structures.

Short time series are sufficient to discriminate orbit types.

Abstract

Recurrence in the phase space of complex systems is a well-studied phenomenon, which has provided deep insights into the nonlinear dynamics of such systems. For dissipative systems, characteristics based on recurrence plots have recently attracted much interest for discriminating qualitatively different types of dynamics in terms of measures of complexity, dynamical invariants, or even structural characteristics of the underlying attractor's geometry in phase space. Here, we demonstrate that the latter approach also provides a corresponding distinction between different co-existing dynamical regimes of the standard map, a paradigmatic example of a low-dimensional conservative system. Specifically, we show that the recently developed approach of recurrence network analysis provides potentially useful geometric characteristics distinguishing between regular and chaotic orbits. We find that chaotic orbits in an intermittent laminar phase (commonly referred to as sticky orbits) have a distinct geometric structure possibly differing in a subtle way from those of regular orbits, which is highlighted by different recurrence network properties obtained from relatively short time series. Thus, this approach can help discriminating regular orbits from laminar phases of chaotic ones, which presents a persistent challenge to many existing chaos detection techniques.