Synchronization and averaging in partially hyperbolic systems with fast and slow variables

Abstract

We study a family of dynamical systems obtained by coupling an Anosov map on the two-dimensional torus -- the chaotic system -- with the identity map on the one-dimensional torus -- the neutral system -- through a dissipative interaction. We show that the two systems synchronize: the trajectories evolve toward an attracting invariant manifold, and the full dynamics is conjugated to its linearization around the invariant manifold. As a byproduct, we obtain that there exists a unique exponentially mixing physical measure. When the interaction is small, the evolution of the variable which describes the neutral system is very close to the identity; hence, it appears as a slow variable with respect to the variable which describes the chaotic system, and which is wherefore named the fast variable. We demonstrate that, seen on a suitably long time scale, the slow variable effectively follows the solution of a deterministic differential equation obtained by averaging over the fast variable. More precisely, we prove that the invariant manifold is in probability close to the fixed point of the averaged dynamics and that the difference between the exact evolution of the slow variable, seen from the invariant manifold, and its averaged evolution is in probability exponentially decreasing for arbitrarily large times.