**Fields Institute Monographs**

Volume: 11;
1999;
245 pp;
Hardcover

MSC: Primary 34;
Secondary 58

Print ISBN: 978-0-8218-1074-3

Product Code: FIM/11

List Price: $89.00

Individual Member Price: $71.20

**Electronic ISBN: 978-1-4704-3138-9
Product Code: FIM/11.E**

List Price: $89.00

Individual Member Price: $71.20

# Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback

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*Tibor Krisztin; Hans-Otto Walther; Jianhong Wu*

A co-publication of the AMS and Fields Institute

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).

#### Table of Contents

# Table of Contents

## Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback

- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- Introduction 18
- The delay differential equation and the hypotheses 916
- The separatrix 1522
- The leading unstable set of the origin 1926
- Oscillation frequencies 2734
- Graph representations 3340
- Dynamics on \overline𝑊 and disk representation of \overline𝑊∩𝑆 4148
- Minimal linear instability of the periodic orbit 𝒪 5158
- Smoothness of 𝑊∩𝑆 in case 𝒪 is hyperbolic 5764
- Smoothness of 𝑊∩𝑆 in case 𝒪 is not hyperbolic 6370
- The unstable set of 𝒪 contains the nonstationary points of bd𝒲 6774
- bd𝑊 contains the unstable set of the periodic orbit 𝒪 7582
- 𝐻∩\overline𝑊 is smooth near 𝑝₀ 95102
- Smoothness of \overline𝑊, bd𝑊 and \overline𝑊∩𝑆 113120
- Homeomorphisms from bd𝑊 onto the sphere and the cylinder 125132
- Homeomorphisms from \overline𝑊 onto the closed ball and the solid cylinder 135142
- Resumé 161168
- Appendix I. Equivalent norms, invariant manifolds, Poincaré maps and asymptotic phases 167174
- Appendix II. Smooth center-stable manifolds for 𝐶¹-maps 173180
- Appendix III. Smooth generalized center-unstable manifolds for 𝐶¹-maps 189196
- Appendix IV. Invariant cones close to neutrally stable fixed points with 1-dimensional center spaces 197204
- Appendix V. Unstable sets of periodic orbits 205212
- Appendix VI. A discrete Lyapunov functional and a-priori estimates 211218
- Appendix VII. Floquet multipliers for a class of linear periodic delay differential equations 221228
- Appendix VIII. Some results from topology 237244
- Bibliography 239246
- Index 243250
- Back Cover Back Cover1253

#### 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