Mode-Locking in Driven Disordered Systems as a Boundary-Value Problem

TL;DR

This paper presents an analytical approach to understanding mode-locking phenomena in disordered media, focusing on the shape of Arnol'd tongues for a driven overdamped system, applicable to various pinning potentials.

Contribution

It introduces an exact analytical method for determining Arnol'd tongue shapes in mode-locking, extending beyond perturbative regimes and applicable to different pinning potentials.

Findings

Exact solutions for scalloped pinning potential

Analytical shape of Arnol'd tongues at low amplitude/high frequency

Complementary to perturbative analysis in weak pinning regimes

Abstract

We study mode-locking in disordered media as a boundary-value problem. Focusing on the simplest class of mode-locking models which consists of a single driven overdamped degree-of-freedom, we develop an analytical method to obtain the shape of the Arnol'd tongues in the regime of low ac-driving amplitude or high ac-driving frequency. The method is exact for a scalloped pinning potential and easily adapted to other pinning potentials. It is complementary to the analysis based on the well-known Shapiro's argument that holds in the perturbative regime of large driving amplitudes or low driving frequency, where the effect of pinning is weak.