Oscillating states of periodically driven anharmonic Langevin systems

TL;DR

This paper studies the long-term behavior of anharmonic Langevin systems under periodic driving, revealing exact solutions and stability of oscillating states using symmetry and Hill equations.

Contribution

It introduces a perturbative scheme leveraging $SL_2$ symmetry to analyze asymptotic distributions in driven anharmonic Langevin systems, providing exact solutions and stability conditions.

Findings

Asymptotic distributions can be exactly determined via Hill equations.

Oscillating states are stable against anharmonic perturbations.

Conditions for existence and periodicity of distributions are established.

Abstract

We investigate the asymptotic distributions of periodically driven anharmonic Langevin systems. Utilizing the underlying $SL_2$ symmetry of the Langevin dynamics, we develop a perturbative scheme in which the effect of periodic driving can be treated nonperturbatively to any order of perturbation in anharmonicity. We spell out the conditions under which the asymptotic distributions exist and are periodic, and show that the distributions can be determined exactly in terms of the solutions of the associated Hill equations. We further find that the oscillating states of these driven systems are stable against anharmonic perturbations.