Synchronization of noisy dissipative systems under discretization

The synchronization of coupled systems is a ubiquitous phenomenon in biological, physical and also social science contexts . In particular, synchronization provides an explanation for the emergence of spontaneous order in the dynamical behavior of coupled systems, which in isolation may exhibit chaotic dynamics. The synchronization of coupled dissipative systems has been investigated mathematically in the case of autonomous systems, both for asymptotically stable equilibria and general attractors, such as chaotic attractors. Analogous results also hold for nonautonomous systems, but require a new concept of a nonautonomous attractor. However, the influence of noise on the synchronization of coupled dissipative systems has been studied only recently. In this paper, we analyze the effects of discretization on a dissipatively coupled system with additive noise. We discretize the system using a drift-implicit Euler scheme with constant step size and show that synchronization of discretized system persists independently of the used step size. In particular, the synchronized discretized system matches the discretized synchronized system, i.e. for the considered dissipatively coupled system with additive noise the order of ‘synchronization’ and ‘discretization’ does not matter. We prove a Theorem for convergence and illustrate our theoretical results by some simulations for a dissipative linear system, showing in details the behavior of the discretized solution