Conditioned Lyapunov exponents for random dynamical systems

TL;DR

This paper introduces conditioned Lyapunov exponents for random dynamical systems to analyze local stability of trajectories confined within bounded domains, especially under unbounded noise, and explores their implications for local bifurcations and synchronization.

Contribution

It develops a new framework of conditioned Lyapunov exponents for stochastic systems, linking local stability analysis with quasi-stationary distributions and bifurcation theory.

Findings

Conditioned Lyapunov exponents characterize local stability within bounded domains.

Negative conditioned Lyapunov exponents imply local synchronization.

A conditioned dichotomy spectrum is introduced and characterized.

Abstract

We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain for asymptotically long times. This is motivated by the desire to characterize local dynamical properties in the presence of unbounded noise (when almost all trajectories are unbounded). We illustrate its use in the analysis of local bifurcations in this context. The theory of conditioned Lyapunov exponents of stochastic differential equations builds on the stochastic analysis of quasi-stationary distributions for killed processes and associated quasi-ergodic distributions. We show that conditioned Lyapunov exponents describe the local stability behaviour of trajectories that remain within a bounded domain and - in particular - that negative conditioned Lyapunov exponents imply local synchronisation. Furthermore, a conditioned dichotomy spectrum is introduced and its main characteristics are established.