Fields Institute Monographs 1999; 245 pp; hardcover Volume: 11 ISBN10: 082181074X ISBN13: 9780821810743 List Price: US$89 Member Price: US$71.20 Order Code: FIM/11
 This book contains recent results about the global dynamics defined by a class of delay differential equations which model basic feedback mechanisms and arise in a variety of applications such as neural networks. The authors describe in detail the geometric structure of a fundamental invariant set, which in special cases is the global attractor, and the asymptotic behavior of solution curves on it. The approach makes use of advanced tools which in recent years have been developed for the investigation of infinitedimensional dynamical systems: local invariant manifolds and inclination lemmas for noninvertible maps, Floquet theory for delay differential equations, a priori estimates controlling the growth and decay of solutions with prescribed oscillation frequency, a discrete Lyapunov functional counting zeros, methods to represent invariant sets as graphs, and PoincaréBendixson techniques for classes of delay differential systems. Several appendices provide the general results needed in the case study, so the presentation is selfcontained. Some of the general results are not available elsewhere, specifically on smooth infinitedimensional centerstable manifolds for maps. Results in the appendices will be useful for future studies of more complicated attractors of delay and partial differential equations. Titles in this series are copublished with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada). Readership Graduate students and research mathematicians working in dynamical systems; mathematical biologists. Reviews "In addition to an impressive array of modern techniques of nonlinear analysis, the book contains a number of appendices which summarize, and in some cases prove for the first time, general analytical results needed in the study. For this reason alone the book is a valuable contribution to the subject."  Mathematical Reviews, Featured Review Table of Contents  Introduction
 The delay differential equation and the hypotheses
 The separatrix
 The leading unstable set of the origin
 Oscillation frequencies
 Graph representations
 Dynamics on \(\overline W\) and disk representation of \(\overline W \cap S\)
 Minimal linear instability of the periodic orbit \(\mathcal O\)
 Smoothness of \(W \cap S\) in case \(\mathcal O\) is hyperbolic
 Smoothness of \(W \cap S\) in case \(\mathcal O\) is not hyperbolic
 The unstable set of \(\mathcal O\) contains the nonstationary points of bd\(W\)
 bd\(W\) contains the unstable set of the periodic orbit \(\mathcal O\)
 \(H \cap \overline W\) is smooth near \(p_0\)
 Smoothness of \(\overline W\), bd\(W\) and \(\overline W \cap S\)
 Homeomorphisms from bd\(W\) onto the sphere and the cylinder
 Homeomorphisms from \(\overline W\) onto the closed ball and the solid cylinder
 Resumé
 Equivalent norms, invariant manifolds, Poincaré maps and asymptotic phases
 Smooth centerstable manifolds for \(C^1\)maps
 Smooth generalized centerunstable manifolds for \(C^1\)maps
 Invariant cones close to neutrally stable fixed points with 1dimensional center spaces
 Unstable sets of periodic orbits
 A discrete Lyapunov functional and apriori estimates
 Floquet multipliers for a class of linear periodic delay differential equations
 Some results from topology
 Bibliography
 Index
