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

Tibor Krisztin University of Szeged, Bolyai Institute, Hungary
Hans-Otto Walther Universität Giessen, Giessen, Germany
Jianhong Wu York University, North York, ON, Canada
A co-publication of the AMS and Fields Institute
Available Formats:
Hardcover ISBN: 978-0-8218-1074-3
Product Code: FIM/11
245 pp
List Price: $95.00 MAA Member Price:$85.50
AMS Member Price: $76.00 Electronic ISBN: 978-1-4704-3138-9 Product Code: FIM/11.E 245 pp List Price:$89.00
MAA Member Price: $80.10 AMS Member Price:$71.20
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AMS Member Price: $114.00 Click above image for expanded view Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback Tibor Krisztin University of Szeged, Bolyai Institute, Hungary Hans-Otto Walther Universität Giessen, Giessen, Germany Jianhong Wu York University, North York, ON, Canada A co-publication of the AMS and Fields Institute Available Formats:  Hardcover ISBN: 978-0-8218-1074-3 Product Code: FIM/11 245 pp  List Price:$95.00 MAA Member Price: $85.50 AMS Member Price:$76.00
 Electronic ISBN: 978-1-4704-3138-9 Product Code: FIM/11.E 245 pp
 List Price: $89.00 MAA Member Price:$80.10 AMS Member Price: $71.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$142.50
MAA Member Price: $128.25 AMS Member Price:$114.00
• Book Details

Fields Institute Monographs
Volume: 111999
MSC: 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 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.

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 center-stable manifolds for $C^1$-maps
• Appendix III. Smooth generalized center-unstable manifolds for $C^1$-maps
• Appendix IV. Invariant cones close to neutrally stable fixed points with 1-dimensional center spaces
• Appendix V. Unstable sets of periodic orbits
• Appendix VI. A discrete Lyapunov functional and a-priori 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
• Request Review Copy
Volume: 111999
MSC: 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 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.

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 center-stable manifolds for $C^1$-maps
• Appendix III. Smooth generalized center-unstable manifolds for $C^1$-maps
• Appendix IV. Invariant cones close to neutrally stable fixed points with 1-dimensional center spaces
• Appendix V. Unstable sets of periodic orbits
• Appendix VI. A discrete Lyapunov functional and a-priori 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
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