On the recurrence and Lyapunov time scales of the motion near the chaos border

TL;DR

This paper investigates the relationship between recurrence time and Lyapunov time near the chaos border, showing a quadratic relationship when Lyapunov exponents are measured over short intervals, which explains previous numerical findings.

Contribution

It establishes conditions under which the recurrence and Lyapunov times are statistically related, particularly near the chaos border, and clarifies the quadratic relationship observed in numerical studies.

Findings

The relationship between $T_r$ and $T_L$ resembles a quadratic form under certain conditions.

Numerical measurements of Lyapunov exponents over short intervals reveal a specific statistical relationship.

The results explain previously observed numerical behaviors in chaotic systems.

Abstract

Conditions for the emergence of a statistical relationship between $T_r$, the chaotic transport (recurrence) time, and $T_L$, the local Lyapunov time (the inverse of the numerically measured largest Lyapunov characteristic exponent), are considered for the motion inside the chaotic layer around the separatrix of a nonlinear resonance. When numerical values of the Lyapunov exponents are measured on a time interval not greater than $T_r$, the relationship is shown to resemble the quadratic one. This tentatively explains numerical results presented in the literature.