Stability of synchronization in dissipatively driven Frenkel-Kontorova models

TL;DR

This paper proves that dissipatively driven Frenkel-Kontorova models tend to synchronize over time, introduces a new Lyapunov function and 2D attractor representation, and explores dynamical phase transitions.

Contribution

It provides a rigorous proof of synchronization in these models, introduces novel analytical tools, and characterizes phase transitions.

Findings

Models asymptotically synchronize for various initial conditions

Introduces a new Lyapunov function for analysis

Characterizes dynamical phase transitions

Abstract

We rigorously show that dissipatively driven Frenkel-Kontorova models with either uniform or time-periodic driving asymptotically synchronize for a wide range of initial conditions. The main tool is a new Lyapunov function, as well as a 2D representation of the attractor. We then characterize dynamical phase transitions and outline new algorithms for determining them.