# Stochastic modelling of dynamical systems with several attractors

## TL;DR

This paper explores stochastic models for systems with multiple attractors, comparing mean-field and Langevin approaches through numerical simulations to better understand their long-term behavior.

## Contribution

It introduces a mean-field approach for finite systems with multiple attractors and compares it with traditional Langevin models using numerical simulations.

## Key findings

- Mean-field hypothesis captures bifurcation behavior in finite systems.
- Numerical simulations show agreement between mean-field and Langevin approaches.
- Long-term dynamics are better understood with the proposed stochastic modelling.

## Abstract

The usual Langevin approach to describe systems driven by noise fails to describe the long time behavior of systems with multiple attractors. The solution of the associated linear Fokker-Planck equation is always unique, even though it might show one or more maxima. In this context, it is customary to call transitions to changes in the shape of the equilibrium distribution function and relate the maxima to the attractors. Some years ago, a theory was developed for a system with interacting elements or subunits that, starting from the Langevin description of all the variables, leads to bifurcating \textquotedblleft one-particle\textquotedblright\, distribution functions when the number of elements tends to infinity. In this paper, a mean-field hypothesis has been used to deal with systems with a finite number of elements. We carry out numerical simulations yielding bifurcation solutions for the probability density of a collective variable. We also compare the results of the mean-field hypothesis with those obtained with the Langevin approach for finite systems.