Synchronization and averaging in a simple dynamical systems with fast/slow variables

Math Physics Seminar
Thursday, February 9, 2023 - 12:00pm for 1 hour (actually 50 minutes)
Skiles Room 005, and online zoom link: Meeting ID: 961 2577 3408
Federico Bonetto – School of Mathematics, Georgia Tech –
Michael Loss

 We study a family of dynamical systems obtained by coupling a chaotic (Anosov) map on the two-dimensional torus -- the chaotic variable -- with the identity map on the one-dimensional torus -- the neutral variable -- through a dissipative interaction. We show that the  two systems synchronize, in the sense that the trajectories evolve toward an attracting invariant manifold, and that the full dynamics is conjugated to its linearization around the invariant manifold. When the interaction is small, the evolution of the neutral variable is very close to the identity and hence the neutral variable appears as a slow variable with respect to the fast chaotic variable: we show 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.

The seminar can also be accessed online via zoom link: Meeting ID: 961 2577 3408