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Synchronization of coupled map lattices

Part of: Dynamical systems with hyperbolic behavior Time-dependent statistical mechanics (dynamic and nonequilibrium) Smooth dynamical systems: general theory Infinite-dimensional dissipative dynamical systems

Published online by Cambridge University Press:  30 March 2023

Alexandre Baraviera ,
Pedro Duarte  and
Maria Joana Torres
Alexandre Baraviera
Affiliation:
Instituto de Matemática e Estatística, UFRGS, Av. Bento Gonçalves 9500, Porto Alegre, RS 91500, Brazil (baravi@mat.ufrgs.br)
Pedro Duarte
Affiliation:
CMAF, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Lisboa 1749-016, Portugal (pmduarte@fc.ul.pt)
Maria Joana Torres
Affiliation:
CMAT and Departamento de Matemática, Universidade do Minho, Campus de Gualtar, Braga 4700-057, Portugal (jtorres@math.uminho.pt)
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Abstract

In this paper, we address the issue of synchronization of coupled systems, introducing concepts of local and global synchronization for a class of systems that extend the model of coupled map lattices. A criterion for local synchronization is given; numerical experiments are exhibited to illustrate the criteria and also to raise some questions in the end of the text.

Keywords

stochastic kernelglobal/local synchronizationcoupled map latticeHardy-Littlewood maximal operatorAmari neural networkMarkov chain

MSC classification

Primary: 37C99: None of the above, but in this section
Secondary: 37D99: None of the above, but in this section 37L60: Lattice dynamics 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 1 , February 2023 , pp. 143 - 163
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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References

Amari, S.-I., Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybernet. 27(2) (1977), 77–87.10.1007/BF00337259CrossRefGoogle ScholarPubMed
Bressloff, P. C., Spatiotemporal dynamics of continuum neural fields, J. Phys. A 45(3) (2012), .10.1088/1751-8113/45/3/033001CrossRefGoogle Scholar
Bunimovich, L. A. and Sinaĭ, Y. G., Spacetime chaos in coupled map lattices, Nonlinearity 1(4) (1988), 491–516.10.1088/0951-7715/1/4/001CrossRefGoogle Scholar
Bunimovich, L., Ming-Chia, L. and Lyu, M.-J., Covering relations for coupled map networks, J. Math. Anal. Appl. 396(1) (2012), 189–198.10.1016/j.jmaa.2012.06.005CrossRefGoogle Scholar
Doob, J. L., Stochastic processes (Wiley, New York, 1990), Reprint of the 1953 original, A Wiley-Interscience Publication.Google Scholar
Duarte, P. and Torres, M. J., Spectral stability of Markov systems, Nonlinearity 21(3) (2008), 381–397.10.1088/0951-7715/21/3/001CrossRefGoogle Scholar
Faranda, D., Ghoudi, H., Guiraud, P. and Vaienti, S., Extreme value theory for synchronization of coupled map lattices, Nonlinearity 31(7) (2018), 3326–3358.10.1088/1361-6544/aabc8eCrossRefGoogle Scholar
Huang, Y. and Feng, Z., Infinite-dimensional dynamical systems induced by interval maps, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13 (2006), 509–524.Google Scholar
Jost, J. and Joy, M. P., Spectral properties and synchronization in coupled map lattices, Phys. Rev. E 65 (2001), .10.1103/PhysRevE.65.016201CrossRefGoogle ScholarPubMed
Keller, G., An ergodic theoretic approach to mean field coupled maps, in Fractal geometry and stochastics II (eds. Bandt, C., Graf, S. and Zähle, M.), (Birkhäuser, Basel, 2000).Google Scholar
Keller, G. and Liverani, C., Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension, Commun. Math. Phys. 262(1) (2006), 33–50.10.1007/s00220-005-1474-7CrossRefGoogle Scholar
Keller, G., Künzle, M. and Nowicki, T., Some phase transitions in coupled map lattices, Phys. D 59(1–3) (1992), 39–51.10.1016/0167-2789(92)90205-2CrossRefGoogle Scholar
Poignard, C., Discrete synchronization of massively connected systems using hierarchical couplings, Phys. D 320 (2016), 19–37.10.1016/j.physd.2015.12.001CrossRefGoogle Scholar
Salasnich, L., Power spectrum and diffusion of the Amari neural field, Symmetry 11(2) (2019), .10.3390/sym11020134CrossRefGoogle Scholar
Schwartz, L., Radon measures on arbitrary topological spaces and cylindrical measures (Oxford University Press, 1973).Google Scholar
Sélley, F. M. and Tanzi, M., Synchronization for networks of globally coupled maps in the thermodynamic limit J. Stat. Phys. 189 (2022), .10.1007/s10955-022-02968-6CrossRefGoogle Scholar