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.
Asymptotic properties of mean field coupled maps
2019. 03. 08. 14:15
Fanni Mincsovicsné Sélley (Rényi Institute and BME MI)