Synchronization and Phase Ordering in Globally Coupled Chaotic Maps
We investigate the processes of synchronization and phase ordering in a system of globally coupled maps possessing bistable, chaotic local dynamics. The stability boundaries of the synchronized states are determined on the space of parameters of the system. The collective properties of the system are characterized by means of the persistence probability of equivalent spin variables that define two phases, and by a magnetization-like order parameter that measures the phase-ordering behavior. As a consequence of the global interaction, the persistence probability saturates for all values of the coupling parameter, in contrast to the transition observed in the temporal behavior of the persistence in coupled maps on regular lattices. A discontinuous transition from a nonordered state to a collective phase-ordered state takes place at a critical value of the coupling. On an interval of the coupling parameter, we find three distinct realizations of the phase-ordered state, which can be discerned by the corresponding values of the saturation persistence. Thus, this statistical quantity can provide information about the transient behaviors that lead to the different phase configurations in the system. The appearance of disordered and phase-ordered states in the globally coupled system can be understood by calculating histograms and the time evolution of local map variables associated to the these collective states.
KeywordsCoupling Strength Coupling Parameter Collective Behavior Spin Variable Chaotic Orbit
This work is supported by project No. C-1906-14-05-B from CDCHTA, Universidad de Los Andes, Venezuela. O. A. thanks Projet Prometeo, Secretaría de Educación Superior, Ciencia, Tecnología e Innovación, Senescyt, Ecuador. M. G. C. is grateful to the Senior Associates Program of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
- 7.Yakovenko, V.M.: Encyclopedia of Complexity and System Science, edited by Meyers, R.A. Springer, New York (2009)Google Scholar
- 11.Manrubia, S.C., Mikhailov, A.S., Zanette, D.H.: Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems. World Scientific, Singapore (2004)Google Scholar