The largest Lyapunov exponent as a tool for detecting relative changes in the particle positions

TL;DR

This paper demonstrates that the largest Lyapunov exponent can detect microchanges in particle configurations within the driven Frenkel-Kontorova model, revealing subtle dynamics not observable through other measures.

Contribution

It introduces the use of the largest Lyapunov exponent for detecting microchanges in chain configurations in the Frenkel-Kontorova model, including both damped and overdamped cases.

Findings

Jumps in the Lyapunov exponent indicate microchanges in particle positions.

The occurrence of jumps depends on the type of commensurate structure and substrate deformation.

Minimal force for collective motion is independent of Lyapunov exponent jumps.

Abstract

Dynamics of the driven Frenkel-Kontorova model with asymmetric deformable substrate potential is examined by analyzing response function, the largest Lyapunov exponent and Poincar\'{e} sections for two neighboring particles. The obtained results show that the largest Lyapunov exponent, besides being used for investigating integral quantities, can be used for detecting microchanges in chain configuration of both damped Frenkel-Kontorova model with inertial term and its strictly overdamped limit. Slight changes in relative positions of the particles are registered through jumps of the largest Lyapunov exponent in the pinning regime. The occurrence of such jumps is highly dependent on type of commensurate structure and deformation of substrate potential. The obtained results also show that the minimal force required to initiate collective motion of the chain is not dependent on the number of Lyapunov exponent jumps in the pinning regime. These jumps are also registered in the sliding regime, where they are a consequence of a more complex structure of largest Lyapunov exponent on the step.