Cyclic feedback systems
About this Title
Publication: Memoirs of the American Mathematical Society
Publication Year 1998: Volume 134, Number 637
ISBNs: 978-0-8218-0783-5 (print); 978-1-4704-0226-6 (online)
MathSciNet review: 1432140
MSC: Primary 34C35; Secondary 34A26, 34C23, 34C37, 58F12, 58F13
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.
Research mathematicians and graduate students interested in the structure of attractors (and repellors); biologists; electrical engineers.
Table of Contents
- 1. Introduction
- 2. Linear theory
- 3. Main results
- 4. Proofs of the lemmas
- 5. Proof of Theorem 1.13