Coupled map lattice

Coupled-map Lattices. Another obvious generalization is to lift the restriction that sites can take on only one of a few discrete values. Coupled-map lattices are CA models in which continuity is restored to the state space. That is to say, the cell values are no longer constrained to take on only the values 0 and 1 as in the examples discussed above, but can now take on arbitrary real values. First introduced by Kaneko [kaneko83]-[kaneko93], such systems are simpler than partial differential equations but more complex than generic CA. Coupled-map lattices are discussed in chapter 8. 

Chapter 8 describes a number of generalized CA models, including reversible CA, coupled-map lattices, quantum CA, reaction-diffusion models, immunologically motivated CA models, random Boolean networks, sandpile models (in the context of self-organized criticality), structurally dynamic CA (in which the temporal evolution of the value of individual sites of a lattice are dynamically linked to an evolving lattice structure), and simple CA models of combat. 

Coupled-Map Lattices these are models in which continuity is restored to the state space ... 

An altogether different behavior emerges for slightly larger values of 7 ( 40 -tSee also our discus.sion of spatiotemporal chaos in coupled-map lattices in section 8.2. 

V0I.670 A. Dinklage, G. Marx, T. Klinger, L. Schweikhard (Eds.), Plasma Physics V0I.671 J.-R. Chazottes, B. Fernandez (Eds.), Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems... 

Coupled lattices of various types can be created. In the cellular automata approach, a variable that can take on discrete values constitutes the elements which are coupled together. The coupling occurs via rules that simulate physical processes such as biological interactions, diffusion, and so forth. Coupled map lattices take this one step further and assign to each lattice element a difference equation that, when iterated, produces a discrete dynamical system. Coupled ODE lattices represent the next step in complexity, and accuracy, for a coupled lattice here, an ODE or system of ODEs are coupled together, again by a choice of simple rules chosen to simulate the desired physical interaaions. 

There are three ways to simulate reaction-diffusion system. The traditional method is to solve partial differential equation directly. Another way is to divide system into cells, which is called cell dynamic scheme (CDS). Typical models are cellular automata (CA)[176] and coupled map lattice (CML)[177]. In cellular automata model, each value of the cell (lattice) is digital. On the other hand, in coupled map lattice model, each value of the lattice (cell) is continuous. CA model is microscopic while CML model is mesoscopic. The advantage of the CML is compatibility with the physical phenomena by smaller number of cells and numerical stability. Therefore, the model based on CML is developed. Each cell has continuum state and the time step is discrete. Generally, each cell is static and not deformable. Deformable cell (lattice) is supposed in order to represent deformation process of the gel. Each cell deforms based on the internal state, which is determined by the reaction between the cell and the environment. 

In contrast to these basic approaches at the macroscopic and mesoscopic levels, one can consider a class of models that does not rely on a knowledge of the detailed rate law or reaction mechanism but instead abstract certain generic features of the behavior. These simplified models often provide insight into the system s dynamics and isolate the minimal features needed to rationalize complex phenomena. Cellular automata (CA) and coupled map lattices (CML) are two examples of such abstract models that we shall discuss. In the following sections, we discuss each of these models, give some of the background that led to their formulation, and provide an introduction to how they are constructed. The presentation will focus on a few examples instead of providing an exhaustive overview. 

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