Application of largest Lyapunov exponent analysis on the studies of dynamics under external forces

TL;DR

This paper demonstrates that the largest Lyapunov exponent is an effective and sensitive tool for analyzing the dynamics of driven dissipative systems, providing insights into depinning forces and Shapiro steps.

Contribution

It introduces the use of the largest Lyapunov exponent analysis as a novel, convenient method to study system dynamics under external forces, especially in detecting depinning and Shapiro steps.

Findings

Largest Lyapunov exponent estimates depinning forces in dc-driven systems.

Lyapunov exponent detects Shapiro steps more sensitively in dc+ac driven systems.

Behavior of Lyapunov exponent mirrors the amplitude dependence of depinning force.

Abstract

Dynamics of driven dissipative Frenkel-Kontorova model is examined by using largest Lyapunov exponent computational technique. Obtained results show that besides the usual way where behavior of the system in the presence of external forces is studied by analyzing its dynamical response function, the largest Lyapunov exponent analysis can represent a very convenient tool to examine system dynamics. In the dc driven systems, the critical depinning force for particular structure could be estimated by computing the largest Lyapunov exponent. In the dc+ac driven systems, if the substrate potential is the standard sinusoidal one, calculation of the largest Lyapunov exponent offers a more sensitive way to detect the presence of Shapiro steps. When the amplitude of the ac force is varied the behavior of the largest Lyapunov exponent in the pinned regime completely reflects the behavior of Shapiro steps and the critical depinning force, in particular, it represents the mirror image of the amplitude dependence of critical depinning force. This points out an advantage of this technique since by calculating the largest Lyapunov exponent in the pinned regime we can get an insight into the dynamics of the system when driving forces are applied.