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 infinite-dimensional 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 Poincaré-Bendixson techniques for classes of delay differential systems.

Several appendices provide the general results needed in the case study, so the presentation is self-contained. Some of the general results are not available elsewhere, specifically on smooth infinite-dimensional center-stable 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 co-published 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 center-stable manifolds for $C^1$-maps
• Smooth generalized center-unstable manifolds for $C^1$-maps
• Invariant cones close to neutrally stable fixed points with 1-dimensional center spaces
• Unstable sets of periodic orbits
• A discrete Lyapunov functional and a-priori estimates
• Floquet multipliers for a class of linear periodic delay differential equations
• Some results from topology
• Bibliography
• Index