Coupled Map Lattices

For description of complex dynamics and chaos in extended systems of different nature (turbulence, electronic devices, nonlinear optics, chemical reaction-diffusion systems, biological populations) a number of model classes may be used: partial differential equations, chains of oscillators, coupled map lattices, cellular automata. If we speak on qualitative understanding of the complex spatio-temporal dynamics, it is sometimes preferable to deal not with a continuous medium, but with lattice models.

Coupled map lattices - CML were introduced in 80-th (K.Kaneko, R.Kapral, S.Kuznetsov). Let us imagine a one- or two-dimensional lattice and assume that a discrete time system, say a quadratic map, is placed in each cell. Next, suppose that each cell is coupled with its neighbors in such way that their states effect the state of the given cell in the next step of time.

Sometimes the use of lattice may be regarded as an approximate approach to description of a continuous medium, then the equations should have a reasonable continuous limit with decrease of the spatial step.

In other cases, the lattice model may be appropriate accounting essence of the problem. For example, in solid state physics a natural discretization appears due to presence of crystal lattice.

Lattice systems may be constructed artificially, e.g. in electronics and optics, to realize devices with novel operational possibilities. (One of the first works where the model of CML type was suggested had arisen from analysis of electronic delay-feedback generator.)

If the individual block element of the lattice is a map demonstrating the period-doubling transition to chaos, then the Feigenbaum regularities of universality and scaling are valid for it. It is natural to ask, how this circumstance will reveal itself in dynamics of coupled systems? Will the transition to spatio-temporal chaos be also associated with some universality and scaling properties?

Saratov group
of theoretical nonlinear
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