Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

Abstract

AbstractThe nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor 𝒜 in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ϵ‐entropy. Upper and lower bounds of this entropy are obtained.Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ℝn. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space 𝒜. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy.In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor.Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.