Electronic ISBN:  9781470400583 
Product Code:  MEMO/100/481.E 
List Price:  $38.00 
MAA Member Price:  $34.20 
AMS Member Price:  $22.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 100; 1992; 179 ppMSC: Primary 58; Secondary 34; 47; 54; 55;
This work is devoted to a detailed study of the equivariant degree and its applications for the case of an \(S^1\)action. This degree is an element of the equivariant homotopy group of spheres, which are computed in a stepbystep extension process. Applications include the index of an isolated orbit, branching and Hopf bifurcation, and period doubling and symmetry breaking for systems of autonomous differential equations. The authors have paid special attention to making the text as selfcontained as possible, so that the only background required is some familiarity with the basic ideas of homotopy theory and of Floquet theory in differential equations. Illustrating in a natural way the interplay between topology and analysis, this book will be of interest to researchers and graduate students.
ReadershipResearchers and graduate students who wish to learn about the interplay between topology and analysis.

Table of Contents

Chapters

1. Preliminaries

2. Extensions of $S^1$maps

3. Homotopy groups of $S^1$maps

4. Degree of $S^1$maps

5. $S^1$index of an isolated nonstationary orbit and applications

6. Index of an isolated orbit of stationary solutions and applications

7. Virtual periods and orbit index

Appendix. Additivity up to one suspension


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This work is devoted to a detailed study of the equivariant degree and its applications for the case of an \(S^1\)action. This degree is an element of the equivariant homotopy group of spheres, which are computed in a stepbystep extension process. Applications include the index of an isolated orbit, branching and Hopf bifurcation, and period doubling and symmetry breaking for systems of autonomous differential equations. The authors have paid special attention to making the text as selfcontained as possible, so that the only background required is some familiarity with the basic ideas of homotopy theory and of Floquet theory in differential equations. Illustrating in a natural way the interplay between topology and analysis, this book will be of interest to researchers and graduate students.
Researchers and graduate students who wish to learn about the interplay between topology and analysis.

Chapters

1. Preliminaries

2. Extensions of $S^1$maps

3. Homotopy groups of $S^1$maps

4. Degree of $S^1$maps

5. $S^1$index of an isolated nonstationary orbit and applications

6. Index of an isolated orbit of stationary solutions and applications

7. Virtual periods and orbit index

Appendix. Additivity up to one suspension