Convergence to local random attractors

Abstract

Random attractors allow one to classify qualitative and quantitative aspects of the long-time behaviour of stochastically forced systems viewed as random dynamical systems (RDS) in an analogous way to attractors for deterministic systems. We compare several notions of random attractor in RDS by examining convergence of trajectories to a random invariant set in different ways, including convergence of mean distance, convergence in probability and convergence on dense subsequences as well as pullback convergence. We give examples showing how these concepts of attraction are inequivalent. We also examine definitions for local attractors and possible methods of decomposition of random attractors into 'random Milnor' attractors. We point out some problems that remain in interpreting these. Finally, we examine numerical simulations of a stochastic van der Pol-Duffing equation and find cases where there appear to be Milnor attractors with positive top Lyapunov exponents. We explain bursting behaviour of the two-point motion for some parameter values in terms of the presence of non-trivial random Milnor attractors.