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Synchronization and Phase Ordering in Globally Coupled Chaotic Maps

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Nonlinear Maps and their Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 112))

Abstract

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.

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Acknowledgement

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.

Author information

Authors and Affiliations

  1. Departamento de Física, FACYT, Universidad de Carabobo, Valencia, Venezuela

    O. Alvarez-Llamoza

  2. Centro de Física Fundamental, Universidad de Los Andes, Mérida, Venezuela

    M. G. Cosenza

  3. Universidad Católica de Cuenca, Cuenca, Ecuado

    O. Alvarez-Llamoza

Authors
  1. O. Alvarez-Llamoza
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  2. M. G. Cosenza
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Corresponding author

Correspondence to O. Alvarez-Llamoza .

Editor information

Editors and Affiliations

  1. Facultad de Ciencias, Universidad de Zaragoza, Zaragoza, Spain

    Ricardo López-Ruiz

  2. Université de Toulouse, LAAS-CNRS, INSA, Toulouse, France

    Danièle Fournier-Prunaret

  3. Department of E.E. Eng., Tokushima University, Tokushima, Japan

    Yoshifumi Nishio

  4. Department of Mathematics, University of Évora, CIMA-UE, Évora, Portugal

    Clara Grácio

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Alvarez-Llamoza, O., Cosenza, M. (2015). Synchronization and Phase Ordering in Globally Coupled Chaotic Maps. In: López-Ruiz, R., Fournier-Prunaret, D., Nishio, Y., Grácio, C. (eds) Nonlinear Maps and their Applications. Springer Proceedings in Mathematics & Statistics, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-12328-8_14

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