Constructing universal pattern formation processes governed by reaction-diffusion systems

For a given connected compact subset $K$ in $mathbb{R}^n$ we construct a smooth map $F$ on $mathbb{R}^{1+n}$ in such a way that the corresponding reaction-diffusion system $u_t=DDelta u+F(u)$ of $n+1$ components $u=(u_0,u_1,dots ,u_n)$, 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. Submitted August 28, 2002. Published October 4, 2002. Math Subject Classifications: 35B40, 70G60, 35Q99 Key Words: Attractor; pattern formation