Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

Abstract

Summary. We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusion u~ = eDUxx + (A + eAt)u + F(u), u R2. These equations (quasinormal forms) describe the behaviour of solutions of the original equation for 8 ~ 0.One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other three equations are considered. Two of these equations have an interesting feature that may be called a sensitivity to small parameters: they contain a new parameter 0(e)---(ae -1/2 rood l) that influences the behaviour Of solutions, but changes infinitely many times when e ~ 0. This does not create problems'in numerical analysis of quasinormal forms, but makes numerical study of the original problem involving e almost impossible.