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Book DetailsGraduate Studies in MathematicsVolume: 85; 2007; 221 ppMSC: Primary 37; 54
Since at least the time of Poisson, mathematicians have pondered the notion of recurrence for differential equations. Solutions that exhibit recurrent behavior provide insight into the behavior of general solutions. In Recurrence and Topology, Alongi and Nelson provide a modern understanding of the subject, using the language and tools of dynamical systems and topology.
Recurrence and Topology develops increasingly more general topological modes of recurrence for dynamical systems beginning with fixed points and concluding with chain recurrent points. For each type of recurrence the text provides detailed examples arising from explicit systems of differential equations; it establishes the general topological properties of the set of recurrent points; and it investigates the possibility of partitioning the set of recurrent points into subsets which are dynamically irreducible. The text includes a discussion of realvalued functions that reflect the structure of the sets of recurrent points and concludes with a thorough treatment of the Fundamental Theorem of Dynamical Systems.
Recurrence and Topology is appropriate for mathematics graduate students, though a wellprepared undergraduate might read most of the text with great benefit.
ReadershipUndergraduate and graduate students interested in dynamical systems and topology.

Table of Contents

Chapters

Chapter 1. Flows

Chapter 2. Recurrent points

Chapter 3. Irreducible sets

Chapter 4. Test functions

Afterword

Appendix A. Discrete dynamical systems

Appendix B. Circle rotations

Appendix C. The Hausdorff metric


Additional Material

Reviews

The book is clear, and generally wellwritten, and could profitably be read by either graduate students or advanced undergraduates. The presentation is extremely detailed.
Mathematical Reviews


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Since at least the time of Poisson, mathematicians have pondered the notion of recurrence for differential equations. Solutions that exhibit recurrent behavior provide insight into the behavior of general solutions. In Recurrence and Topology, Alongi and Nelson provide a modern understanding of the subject, using the language and tools of dynamical systems and topology.
Recurrence and Topology develops increasingly more general topological modes of recurrence for dynamical systems beginning with fixed points and concluding with chain recurrent points. For each type of recurrence the text provides detailed examples arising from explicit systems of differential equations; it establishes the general topological properties of the set of recurrent points; and it investigates the possibility of partitioning the set of recurrent points into subsets which are dynamically irreducible. The text includes a discussion of realvalued functions that reflect the structure of the sets of recurrent points and concludes with a thorough treatment of the Fundamental Theorem of Dynamical Systems.
Recurrence and Topology is appropriate for mathematics graduate students, though a wellprepared undergraduate might read most of the text with great benefit.
Undergraduate and graduate students interested in dynamical systems and topology.

Chapters

Chapter 1. Flows

Chapter 2. Recurrent points

Chapter 3. Irreducible sets

Chapter 4. Test functions

Afterword

Appendix A. Discrete dynamical systems

Appendix B. Circle rotations

Appendix C. The Hausdorff metric

The book is clear, and generally wellwritten, and could profitably be read by either graduate students or advanced undergraduates. The presentation is extremely detailed.
Mathematical Reviews