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Product Code:  FIM/11 
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Electronic ISBN:  9781470431389 
Product Code:  FIM/11.E 
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Book DetailsFields Institute MonographsVolume: 11; 1999; 245 ppMSC: Primary 34; Secondary 58;
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.ReadershipGraduate students and research mathematicians working in dynamical systems; mathematical biologists.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. The delay differential equation and the hypotheses

Chapter 3. The separatrix

Chapter 4. The leading unstable set of the origin

Chapter 5. Oscillation frequencies

Chapter 6. Graph representations

Chapter 7. Dynamics on $\overline W$ and disk representation of $\overline W \cap S$

Chapter 8. Minimal linear instability of the periodic orbit $\mathcal O$

Chapter 9. Smoothness of $W \cap S$ in case $\mathcal O$ is hyperbolic

Chapter 10. Smoothness of $W \cap S$ in case $\mathcal O$ is not hyperbolic

Chapter 11. The unstable set of $\mathcal O$ contains the nonstationary points of bd$W$

Chapter 12. bd$W$ contains the unstable set of the periodic orbit $\mathcal O$

Chapter 13. $H \cap \overline W$ is smooth near $p_0$

Chapter 14. Smoothness of $\overline W$, bd$W$ and $\overline W \cap S$

Chapter 15. Homeomorphisms from bd$W$ onto the sphere and the cylinder

Chapter 16. Homeomorphisms from $\overline W$ onto the closed ball and the solid cylinder

Chapter 17. Resumé

Appendix I. Equivalent norms, invariant manifolds, Poincaré maps and asymptotic phases

Appendix II. Smooth centerstable manifolds for $C^1$maps

Appendix III. Smooth generalized centerunstable manifolds for $C^1$maps

Appendix IV. Invariant cones close to neutrally stable fixed points with 1dimensional center spaces

Appendix V. Unstable sets of periodic orbits

Appendix VI. A discrete Lyapunov functional and apriori estimates

Appendix VII. Floquet multipliers for a class of linear periodic delay differential equations

Appendix VIII. Some results from topology


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


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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.
Graduate students and research mathematicians working in dynamical systems; mathematical biologists.

Chapters

Chapter 1. Introduction

Chapter 2. The delay differential equation and the hypotheses

Chapter 3. The separatrix

Chapter 4. The leading unstable set of the origin

Chapter 5. Oscillation frequencies

Chapter 6. Graph representations

Chapter 7. Dynamics on $\overline W$ and disk representation of $\overline W \cap S$

Chapter 8. Minimal linear instability of the periodic orbit $\mathcal O$

Chapter 9. Smoothness of $W \cap S$ in case $\mathcal O$ is hyperbolic

Chapter 10. Smoothness of $W \cap S$ in case $\mathcal O$ is not hyperbolic

Chapter 11. The unstable set of $\mathcal O$ contains the nonstationary points of bd$W$

Chapter 12. bd$W$ contains the unstable set of the periodic orbit $\mathcal O$

Chapter 13. $H \cap \overline W$ is smooth near $p_0$

Chapter 14. Smoothness of $\overline W$, bd$W$ and $\overline W \cap S$

Chapter 15. Homeomorphisms from bd$W$ onto the sphere and the cylinder

Chapter 16. Homeomorphisms from $\overline W$ onto the closed ball and the solid cylinder

Chapter 17. Resumé

Appendix I. Equivalent norms, invariant manifolds, Poincaré maps and asymptotic phases

Appendix II. Smooth centerstable manifolds for $C^1$maps

Appendix III. Smooth generalized centerunstable manifolds for $C^1$maps

Appendix IV. Invariant cones close to neutrally stable fixed points with 1dimensional center spaces

Appendix V. Unstable sets of periodic orbits

Appendix VI. A discrete Lyapunov functional and apriori estimates

Appendix VII. Floquet multipliers for a class of linear periodic delay differential equations

Appendix VIII. Some results from topology

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