From collective periodic running states to completely chaotic synchronised states in coupled particle dynamics

TL;DR

This paper studies the transition from collective periodic running states to fully chaotic synchronized states in coupled particles within a periodic potential, analyzing chaos, synchronization, and directed transport phenomena.

Contribution

It introduces a combined analytical and numerical approach to characterize chaos, synchronization, and transport in coupled particle systems with periodic driving.

Findings

Chaotic motion occurs in higher-dimensional phase space with weak coupling.

Strong coupling leads to complete synchronization, eliminating hyperchaos.

Coordinated energy exchange enables particles to overcome potential barriers, resulting in directed transport.

Abstract

We consider the damped and driven dynamics of two interacting particles evolving in a symmetric and spatially periodic potential. The latter is exerted to a time-periodic modulation of its inclination. Our interest is twofold: Firstly we deal with the issue of chaotic motion in the higher-dimensional phase space. To this end a homoclinic Melnikov analysis is utilised assuring the presence of transverse homoclinic orbits and homoclinic bifurcations for weak coupling allowing also for the emergence of hyperchaos. In contrast, we also prove that the time evolution of the two coupled particles attains a completely synchronised (chaotic) state for strong enough coupling between them. The resulting `freezing of dimensionality' rules out the occurrence of hyperchaos. Secondly we address coherent collective particle transport provided by regular periodic motion. A subharmonic Melnikov analysis is utilised to investigate persistence of periodic orbits. For directed particle transport mediated by rotating periodic motion we present exact results regarding the collective character of the running solutions entailing the emergence of a current. We show that coordinated energy exchange between the particles takes place in such a manner that they are enabled to overcome - one particle followed by the other - consecutive barriers of the periodic potential resulting in collective directed motion.