Conditioned Lyapunov exponents for random dynamical systems

• DOI: 10.1090/tran/7803
• Publication Date: 2019-02-27T14:32:09Z
• Authors: Maximilian Engel; Jeroen S. W. Lamb; Martin Rasmussen
• Keywords: ATTRACTORS; Science & Technology; General Mathematics; 37H15; Dynamical Systems (math.DS); 37H10; 01 natural sciences; 510; 0101 Pure Mathematics; QUASI-STATIONARY DISTRIBUTIONS; 0102 Applied Mathematics; Physical Sciences; 37A50, 37H10, 37H15, 60F99; FOS: Mathematics; HOPF-BIFURCATION; 37A50; Mathematics - Dynamical Systems; 0101 mathematics; 60F99; Mathematics; math.DS

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.