Lyapunov exponent of random dynamical systems on the circle

Rotation Numbers for Random Dynamical Systems on the Circle

In this paper, we study rotation numbers of random dynamical systems on the circle. We prove the existence of rotation numbers and the continuous dependence of rotation numbers on the systems. As an application, we prove a theorem on analytic conjugacy to a circle rotation.

Lyapunov Exponent for Products of Random Matrices

Positive Lyapunov Exponent by a Random Perturbation

We study the effect of a random perturbation on a one-parameter family of dynamical systems whose behavior in the absence of perturbation is ill understood. We provide conditions under which the perturbed system is ergodic and admits a positive Lyapunov exponent, with an explicit lower bound, for a large and controlled set of parameter values. Acknowledgements. The authors are indebted to Lai-S...

Statistics of the Lyapunov Exponent in 1D Random Periodic-on-Average Systems

By means of Monte Carlo simulations we show that there are two qualitatively different modes of localization of classical waves in 1D random periodic-on-average systems. States from pass bands and band edges of the underlying band structure demonstrate single parameter scaling with universal behavior. States from the interior of the band gaps do not have universal behavior and require two param...

Evaluation of the Lyapunov Exponent for Stochastic Dynamical Systems with Event Synchronization

We consider stochastic dynamical systems operating under synchronization constraints on system events. The system dynamics is represented by a linear vector equation in an idempotent semiring through second-order state transition matrices with both random and constant entries. As the performance measure of interest, the Lyapunov exponent defined as the asymptotic mean growth rate of the system ...