Dissipative synchronization of nonautonomous and random systems

The synchronization of coupled systems in its many different forms is an ubiquitous phenomenon in biological, physical and even social sciences. In this article we focus attention on the synchronization of coupled dissipative systems. The aim of this article is to introduce the reader to recent results on the dissipative synchronization of nonanutonomous and random dynamical systems as well as to new mathematical ideas and tools from the theories of nonautonomous and random dynamical systems needed to investigate them. We consider mostly systems satisfying a one-sided dissipative Lipschitz condition since their attractors have a simpler structure and results are easier to formulate and illustrate. Essentially, synchronization persists in the presence of environmental (i.e., additive) noise provided asymptotically stable stochastic stationary solutions are used instead of the equilibria points of the corresponding deterministic systems. For systems with multiplicative noise the situation is somewhat more complicated and the synchronization is obtained when the effect of different driving noises is factored out. The synchronization effect is preserved under discretization of the differential equation. In particular, the implicit Euler scheme with constant step size Δt > 0 applied to the coupled system of ODEs generates a discrete time autonomous dynamical system without requiring restrictions on the step size Δt with respect to the synchronization parameter ν, which becomes synchronized to the autonomous system obtained by applying the implicit Euler scheme with constant step size Δt > 0 to the averaged ODE . The effects of discretization on the synchronization of dissipative systems with additive noise is investigated , and its is shown that synchronization effect was preserved using the drift-implicit Euler-Maruyama scheme with constant step size Δt > 0.