Overview
Study of dynamical systems usually concentrates on the properties and the structure of invariant sets, since the understanding of these is the first step in describing the long time behavior of orbits of the entire dynamical system. There are two different sets of problems related to the study of dynamical systems. One, the study of the dynamics in the neighborhood of the critical elements like fixed points or periodic orbits, is relatively well understood. This volume tackles the second set of problems, related to a global dynamics and the global bifurcations. In this volume the author studies dynamics of cyclic feedback systems. The global dynamics is described by a Morse decomposition of the global attractor, defined with the help of a discrete Lyapunov function. The author shows that the dynamics inside individual Morse sets may be very complicated. A three-dimensional system of ODEs with two linear equations is constructed, such that the invariant set is at least as complicated as a suspension of a full shift on two symbols. The questions posed are perhaps as significant as the reported results.