Some physical models described by the reaction-diffusion equation, and coupled map lattices. (English. Russian original) Zbl 1160.37423

Pesin, Ya. B.; Yurchenko, A. A.

Russ. Math. Surv. 59, No. 3, 481-513 (2004); translation from Usp. Mat. Nauk 59, No. 3, 81-114 (2004).

Summary: A number of models are surveyed which appear in physics, biology, chemistry, and other areas and which are described by a reaction-diffusion equation. The corresponding coupled map lattice (CML) system is obtained by discretizing this equation. These CMLs are classified by the type of the dynamics of the local map. Several different types of behavior are observed: Morse-Smale type systems, systems with attractors, and systems with Smale horseshoes.

MSC:

37L60	Lattice dynamics and infinite-dimensional dissipative dynamical systems
35K57	Reaction-diffusion equations
37D15	Morse-Smale systems
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D25	Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37E05	Dynamical systems involving maps of the interval
37N20	Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
92E20	Classical flows, reactions, etc. in chemistry

Keywords:

nonlinear reaction-diffusion equation; coupled map lattices; dynamics of local maps; one-dimensional maps; FitzHugh-Nagumo equation; Morse-Smale type systems