A description of the global attractor for a class of reaction-diffusion systems with periodic solutions

We examine the autonomous reaction-diffusion system [GRAPHICS] with Dirichlet boundary conditions on (0, 1), where alpha, beta are real, a lpha > 0, and g is C-1 and satisfies some conditions which we need in order to prove the existence of solutions. We construct a solution of (RD) for every initial value in L-2((0, 1)) X L- 2((0, 1)), we show that this solution is uniquely determined and that the s olution has C-infinity-smooth representatives for ail positive t. We determ ine the long time behaviour of each solution. In particular, we show that e ach solution of (RD) tends either to the zero solution or to a periodic orb it. We construct all periodic orbits and show that their number is always f inite. It turns out that the global attractor is a finite union of subsets of L-2 X L-2, which are finite-dimensional manifolds, and the dynamics in these se ts can be described completely