New chaos indicators for systems with extremely small Lyapunov exponents

TL;DR

This paper introduces new chaos indicators capable of detecting transitions between diffusion regimes and identifying chaos in systems with near-zero Lyapunov exponents, enhancing analysis of complex dynamical systems.

Contribution

The paper presents novel chaos indicators specifically designed for systems with extremely small Lyapunov exponents, enabling detection of diffusion regime transitions and chaos in zero-exponent systems.

Findings

Successfully detects transition between Arnold and Chirikov diffusion regimes

Identifies chaoticity in systems with zero Lyapunov exponent

Characterizes sub-exponential diffusions effectively

Abstract

We propose new chaos indicators for systems with extremely small positive Lyapunov exponents. These chaos indicators can firstly detect a sharp transition between the Arnold diffusion regime and the Chirikov diffusion regime of the Froeschl\'e map and secondly detect chaoticity in systems with zero Lyapunov exponent such as the Boole transformation and the $S$-unimodal function to characterize sub-exponential diffusions.