Conditioned Lyapunov exponents for random dynamical systems

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.

Cited by in corpus (8)

• Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems
• Noise-induced chaos: a conditioned random dynamics perspective
• Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise
• On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape
• Transition to anomalous dynamics in a simple random map
• Existence and uniqueness of quasi-stationary and quasi-ergodic measures for absorbing Markov chains: a Banach lattice approach
• A General View on Double Limits in Differential Equations
• The conditioned Lyapunov spectrum for random dynamical systems