Convergence to periodic solutions in autonomous reaction-diffusion systems in one space dimension

We consider the autonomous reaction-diffusion system (d)/(dt) ((u)(v)) = lambda Delta((u)(v)) + f((u)(v)) + ((-v)(u)), t > 0, 0 < x < 1, with Dirichlet boundary conditions where f is an element of C-1{R-2, R) and lambda is positive. We show a Poincare-Bendixson theorem, which means that all solutions converge either to zero or to a periodic orbit. Furthermore, we determine stability properties of the periodic orbits. (C) 2001 Academic Press.