Leveraging temporal features of the divergence quantifier of recurrence plot to detect chaos in conservative systems

TL;DR

This paper demonstrates that the divergence quantifier of recurrence plots ($DIV$), traditionally used for dissipative systems, is effective for detecting chaos in conservative systems like the standard map, showing strong agreement with Lyapunov indicators.

Contribution

It introduces the use of $DIV$ as a finite-time chaos indicator for conservative dynamics and compares its performance with the fast Lyapunov indicator (FLI).

Findings

$DIV$ correlates well with FLI in detecting chaos.

Distinct power-law decay rates of $DIV$ are observed for regular and chaotic regions.

$DIV$ provides a computationally efficient method for chaos detection in conservative systems.

Abstract

The recurrence-based divergence quantifier ($DIV$), traditionally applied to dissipative systems, is shown here to be an effective finite-time chaos indicator for conservative dynamics. We benchmark its performances against the well-established fast Lyapunov indicator (FLI), focusing on the standard map, a canonical model of Hamiltonian chaos. Through extensive numerical simulations on moderately long orbits, we find strong agreement between $DIV$ and FLI, supporting the reported correlation between the divergence of recurrences and positive Lyapunov exponents. Additionally, our study sheds more light into asymptotic time properties of $DIV$ by revealing distinct power laws on regular and chaotic components, both in the original and reconstructed phase spaces. In particular, on a regular component, the space average of $DIV$ decays with the time $N$ as $1/N$, mirroring the decay rate of the maximal Lyapunov exponent. On chaotic components, the space average of $DIV$ decreases at a much slower rate, close to $1/\sqrt{N}$. This scaling insight opens new avenues for characterizing chaos from time series. Our numerical results thus demonstrate $DIV$ to be a computationally viable and theoretically rich tool for chaos detection in conservative systems.