Map Lattices

In the preceding two sections, we discussed patterns in chemically reacting media using PDE reaction-diffusion models, where space, time, and chemical concentrations were continuous variables, and cellular automata, where space, time, and the state of a cell were discrete. We now turn our attention to coupled map lattices another type of model that has been used 

At this point it is useful to make comparisons to the Euler solution of the reaction-diffusion equation. If we measure time in units of At so that t/At — t, we can write Eq. [2] (dropping the subscript X) as, 

If this were the only context in which CML models were used, their utility would be severely limited. For values y beyond the stability limit, the Euler method fails and one obtains solutions that fail to represent the solutions of the reaction-diffusion equation. However, it is precisely the rich pattern formation observed in CML models beyond the stability limit that has attracted researchers to study these models in great detail. Coupled map models show spatiotemporal intermittency, chaos, clustering, and a wide range of pattern formation processes. Many of these complicated phenomena can be studied in detail using CML models because of their simplicity and, if there are generic aspects to the phenomena, for example, certain scaling properties, then these could be carried over to real systems in other parameter regimes. The CML models have been used to study chemical pattern formation in bistable, excitable, and oscillatory media.  

One example will serve to illustrate the use of CML models in this more general context. Suppose we consider the BZ reaction, not in the excitable regime as earlier, but in the oscillatory regime. Certain versions of the BZ 

What has all of this to do with coupled map lattices Suppose we take a CML model where the local map at a site leads to cycling among three states a period-3 map. An example is 

As mentioned above, CMLs are simple generalizations of generic CA systems. Confining ourselves for the time being to one-dimension for simplicity, we begin with a one-dimensional lattice of real-valued variables ai t) R whose temporal evolution is given by 

More general models couple the action of a given function at the center site to other sites with another (possibly different) coupling dynamics and individual coupling constants. For example, one generic form is given by 

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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. 

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