Structure of the global attractor of cyclic feedback systems (Q1347233)

The cyclic feedback systems considered in this paper have the form (1) \(\dot x_ 1 = \pm f_ 1(x_ 1, x_ n)\), \(\dot x_ 2 = f_ 2(x_ 2, x_ 1), \dots, \dot x_ n = f_ n(x_ n, x_{n - 1})\). Assuming that the system has a global compact attractor \({\mathcal A}\), the authors are interested in studying the dynamics of (1) on \({\mathcal A}\). More precisely, they give a description of the set of connecting orbits and Morse sets which can occur in a Morse decomposition of \({\mathcal A}\). The appropriate decomposition of \({\mathcal A}\) is suggested by an integer valued map (constant on the orthants of \(\mathbb{R}^ n)\) which is called a Lyapunov function. Indeed, its values cannot increase along the solutions of (1). Under mild assumptions, the system is shown to be equivalent (in the sense of semi-conjugacy) to a simple model with Morse-Smale dynamics. Weaker results are given for the general case.