Structure of the global attractor of cyclic feedback systems

We characterize the dynamics on global attractors of cyclic feedback systems. Under mild restrictions the description is given in terms of a semiconjugacy to a simple model system which possesses Morse-Smale dynamics. However, for the completely general case, no simple model system is feasible and hence we introduce a weaker notion of equivalence, namely, topological semiequivalency. We then prove that the global attractor of a cyclic feedback system is topologically semiequivalent to the original model flow. Main ingredients in the proof are the discrete Lyapunov function introduced by Mallet-Paret and Smith and the Conley index theory.