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A coupled map lattice (CML) is a dynamical system that models the behavior of non-linear 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.〔Kaneko, Kunihiko. "Overview of Coupled Map Lattices." Chaos 2, Num3(1992): 279.〕 Features of the CML are discrete time dynamics, discrete underlying spaces (lattices or networks), and real (number or vector), local, continuous state variables.〔Chazottes, Jean-René, and Bastien Fernandez. Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems. Springer, 2004. pgs 1–4〕 Studied systems include populations, chemical reactions, convection, fluid flow and biological networks. More recently, CMLs have been applied to computational networks 〔Xu, Jian. Wang, Xioa Fan. " Cascading failures in scale-free coupled map lattices." IEEE International Symposium on Circuits and Systems “ ISCAS Volume 4, (2005): 3395–3398.〕 identifying detrimental attack methods and cascading failures. CML’s are comparable to cellular automata models in terms of their discrete features.〔R. Badii and A. Politi, Complexity: Hierarchical Structures and Scaling in Physics (Cambridge University Press,Cambridge, England, 1997).〕 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. ==Introduction== A CML generally incorporates a system of equations (coupled or uncoupled), a finite number of variables, a global or local coupling scheme and the corresponding coupling terms. The underlying lattice can exist in infinite dimensions. Mappings of interest in CMLs generally demonstrate chaotic behavior. Such maps can be found here: List of chaotic maps. A logistic mapping demonstrates chaotic behavior, easily identifiable in one dimension for parameter r > 3.57 (see Logistic map). It is graphed across a small lattice and decoupled with respect to neighboring sites. The recurrence equation is homogeneous, albeit randomly seeded. The parameter r is updated every time step (see Figure 1, Enlarge, Summary): : The result is a raw form of chaotic behavior in a map lattice. The range of the function is bounded so similar contours through the lattice is expected. However, there are no significant spatial correlations or pertinent fronts to the chaotic behavior. No obvious order is apparent. For a basic coupling, we consider a 'single neighbor' coupling where the value at any given site is mapped recursively with respect to itself and the neighboring site . The coupling parameter is equally weighted. : Even though each native recursion is chaotic, a more solid form develops in the evolution. Elongated convective spaces persist throughout the lattice (see Figure 2). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「coupled map lattice」の詳細全文を読む スポンサード リンク
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