Coupled map lattice

A coupled map lattice (CML) is a dynamical system that models the behavior of nonlinear^{[disambiguation needed]} systems (especially partial differential equations). They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems. This includes the dynamics of spatiotemporal chaos where the number of effective degrees of freedom diverges as the size of the system increases.^[1]

Features of the CML are discrete time dynamics, discrete underlying spaces (lattices or networks), and real (number or vector), local, continuous state variables.^[2] Studied systems include populations, chemical reactions, convection, fluid flow and biological networks. More recently, CMLs have been applied to computational networks ^[3] identifying detrimental attack methods and cascading failures.

CMLs are comparable to cellular automata models in terms of their discrete features.^[4] However, the value of each site in a cellular automata network is strictly dependent on its neighbor(s) from the previous time step. Each site of the CML is only dependent upon its neighbors relative to the coupling term in the recurrence equation. However, the similarities can be compounded when considering multi-component dynamical systems.

^ Kaneko, Kunihiko (1992). "Overview of coupled map lattices". Chaos: An Interdisciplinary Journal of Nonlinear Science. 2 (3). AIP Publishing: 279–282. Bibcode:1992Chaos...2..279K. doi:10.1063/1.165869. ISSN 1054-1500. PMID 12779975.
^ Chazottes, Jean-René; Fernandez, Bastien, eds. (2005). Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems. Lecture notes in physics 671. Berlin: Springer. pp. 1–4. ISBN 9783540242895. OCLC 61030071.
^ Xu, Jian; Wang, Xioa Fan (2005). "Cascading Failures in Scale-Free Coupled Map Lattices". 2005 IEEE International Symposium on Circuits and Systems. Vol. 4. pp. 3395–3398. doi:10.1109/ISCAS.2005.1465357. ISBN 0-7803-8834-8.
^ Badii, R.; Politi, A. (1997). Complexity: Hierarchical Structures and Scaling in Physics. Cambridge nonlinear science series 6. Cambridge, England: Cambridge University Press. ISBN 9780521418904. OCLC 34677022.

[1] Kaneko, Kunihiko (1992). "Overview of coupled map lattices". Chaos: An Interdisciplinary Journal of Nonlinear Science. 2 (3). AIP Publishing: 279–282. Bibcode:1992Chaos...2..279K. doi:10.1063/1.165869. ISSN 1054-1500. PMID 12779975.

[chazottes-2] Chazottes, Jean-René; Fernandez, Bastien, eds. (2005). Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems. Lecture notes in physics 671. Berlin: Springer. pp. 1–4. ISBN 9783540242895. OCLC 61030071.

[3] Xu, Jian; Wang, Xioa Fan (2005). "Cascading Failures in Scale-Free Coupled Map Lattices". 2005 IEEE International Symposium on Circuits and Systems. Vol. 4. pp. 3395–3398. doi:10.1109/ISCAS.2005.1465357. ISBN 0-7803-8834-8.

[4] Badii, R.; Politi, A. (1997). Complexity: Hierarchical Structures and Scaling in Physics. Cambridge nonlinear science series 6. Cambridge, England: Cambridge University Press. ISBN 9780521418904. OCLC 34677022.

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