# Stationary states for underdamped anharmonic oscillators driven by Cauchy noise

## TL;DR

This paper investigates how stationary states of underdamped anharmonic oscillators driven by Cauchy noise depend on potential shape and damping, revealing conditions for multimodality and the influence of potential composition.

## Contribution

It provides a detailed analysis of stationary state shapes in underdamped anharmonic oscillators under Cauchy noise, highlighting the effects of damping and potential type on modality.

## Key findings

- Stationary states can be multimodal or unimodal depending on damping and potential.
- Parabolic potential yields a unimodal stationary density described by a 2D alpha-stable distribution.
- Mixtures of potentials can produce bimodal stationary states, which can be destroyed by strong parabolic addition.

## Abstract

Using methods of stochastic dynamics, we have studied stationary states in the underdamped anharmonic stochastic oscillators driven by Cauchy noise. Shape of stationary states depend both on the potential type and the damping. If the damping is strong enough, for potential wells which in the overdamped regime produce multimodal stationary states, stationary states in the underdamped regime can be multimodal with the same number of modes like in the overdamped regime. For the parabolic potential, the stationary density is always unimodal and it is given by the two dimensional $\alpha$-stable density. For the mixture of quartic and parabolic single-well potentials the stationary density can be bimodal. Nevertheless, the parabolic addition, which is strong enough, can destroy bimodlity of the stationary state.