Detecting random bifurcations via rigorous enclosures of large deviations rate functions

TL;DR

This paper develops a rigorous computational method to analyze large deviations rate functions, enabling detection of bifurcations in stochastic systems through eigenvalue estimates of tilted semigroups.

Contribution

It introduces a computer-assisted approach to accurately compute eigenvalues and rate functions for bifurcation analysis in stochastic differential equations.

Findings

Successfully identified bifurcation points in case studies

Demonstrated the transition to positive Lyapunov exponents

Provided a new computational framework for large deviations analysis

Abstract

The main goal of this work is to provide a description of transitions from uniform to non-uniform snychronization in diffusions based on large deviation estimates for finite time Lyapunov exponents. These can be characterized in terms of moment Lyapunov exponents which are principal eigenvalues of the generator of the tilted (Feynman-Kac) semigroup. Using a computer assisted proof, we demonstrate how to determine these eigenvalues and investigate the rate function which is the Legendre-Fenichel transform of the moment Lyapunov function. We apply our results to two case studies: the pitchfork bifurcation and a two-dimensional toy model, also considering the transition to a positive asymptotic Lyapunov exponent.