Asymptotic properties of mean field coupled maps

2019. 03. 08. 14:15
Fanni Mincsovicsné Sélley (Rényi Institute and BME MI)

We study mean field coupled map systems of uniformly expanding circle maps. We first consider N globally coupled doubling maps of the circle with diffusive coupling. Reconsidering and extending the results of Fernandez we prove ergodicity breaking for N=3 and N=4 and showcase some synchronization phenomena for various values of N in case of strong coupling. We then introduce the continuum limit of the system, where we generalize the doubling map to a smooth uniformly expanding circle map T. Now the state of the system is described by a density function and the evolution of an initial density with respect to the transfer operator of the coupled dynamics is studied. We show that for weak enough coupling, a unique, asymptotically stable invariant density exists in a suitable function space. Furthermore, we show that this invariant density depends Lipschitz continuously on the coupling parameter. For sufficiently strong coupling, we prove convergence to a point mass which can be interpreted as chaotic synchronization. To conclude, we provide some outlook on the case of discontinuous T.