The Hopf Bifurcation and Its Applications

  • J. E. Marsden
  • M. McCracken

Part of the Applied Mathematical Sciences book series (AMS, volume 19)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. J. E. Marsden, M. McCracken
    Pages 27-49
  3. J. E. Marsden, M. McCracken
    Pages 50-55
  4. J. E. Marsden, M. McCracken
    Pages 56-62
  5. J. E. Marsden, M. McCracken
    Pages 63-84
  6. J. E. Marsden, M. McCracken
    Pages 85-90
  7. J. E. Marsden, M. McCracken
    Pages 91-94
  8. J. E. Marsden, M. McCracken
    Pages 104-130
  9. J. E. Marsden, M. McCracken
    Pages 131-135
  10. J. E. Marsden, M. McCracken
    Pages 136-150
  11. S. Chow, J. Mallet-Paret
    Pages 151-162
  12. L. N. Howard, N. Kopell
    Pages 163-193
  13. L. N. Howard, N. Kopell
    Pages 194-205
  14. J. E. Marsden, M. McCracken
    Pages 206-218
  15. J. E. Marsden, M. McCracken
    Pages 219-223
  16. Steve Schecter
    Pages 224-229
  17. J. E. Marsden, M. McCracken
    Pages 250-257
  18. J. E. Marsden, M. McCracken
    Pages 258-284

About this book

Introduction

The goal of these notes is to give a reasonahly com­ plete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to spe­ cific problems, including stability calculations. Historical­ ly, the subject had its origins in the works of Poincare [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930. Hopf's basic paper [1] appeared in 1942. Although the term "Poincare­ Andronov-Hopf bifurcation" is more accurate (sometimes Friedrichs is also included), the name "Hopf Bifurcation" seems more common, so we have used it. Hopf's crucial contribution was the extension from two dimensions to higher dimensions. The principal technique employed in the body of the text is that of invariant manifolds. The method of Ruelle­ Takens [1] is followed, with details, examples and proofs added. Several parts of the exposition in the main text come from papers of P. Chernoff, J. Dorroh, O. Lanford and F. Weissler to whom we are grateful. The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations: see for example, Hale [1,2] and Hartman [1]. Of course, other methods are also available. In an attempt to keep the picture balanced, we have included samples of alternative approaches. Specifically, we have included a translation (by L. Howard and N. Kopell) of Hopf's original (and generally unavailable) paper.

Keywords

Bifurcation Calculation Invariant Manifold Morphism dynamical systems equation proof theorem

Authors and affiliations

  • J. E. Marsden
    • 1
  • M. McCracken
    • 2
  1. 1.Department of MathematicsUniversity of California at BerkeleyUSA
  2. 2.Department of MathematicsUniversity of California at Santa CruzUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-6374-6
  • Copyright Information Springer-Verlag New York 1976
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-90200-5
  • Online ISBN 978-1-4612-6374-6
  • Series Print ISSN 0066-5452
  • About this book