Constructing Universal Pattern Formation Processes Governed by Reaction-Diffusion Systems

Abstract

For a given connected compact subset K in ℝn we construct a smooth map F on ℝ1+n in such a way that the corresponding reaction-diffusion system ut = DΔu + F(u) of n + 1 components u = (u0, u1,..., un), accompanying with the homogeneous Neumann boundary condition, has an attractor which is isomorphic to K. This implies the following universality: The make-up of a pattern with arbitrary complexity (e.g., a fractal pattern) can be realized by a reaction-diffusion system once the vector supply term F has been previously properly constructed.