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Applied Mathematical Sciences

The Hopf Bifurcation and Its Applications

Authors: Marsden, J. E., McCracken, M.

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About this book

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.

Table of contents (25 chapters)

Table of contents (25 chapters)

Buy this book

eBook $69.99
price for USA in USD (gross)
  • ISBN 978-1-4612-6374-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $89.99
price for USA in USD
  • ISBN 978-0-387-90200-5
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
The Hopf Bifurcation and Its Applications
Authors
Series Title
Applied Mathematical Sciences
Series Volume
19
Copyright
1976
Publisher
Springer-Verlag New York
Copyright Holder
Springer-Verlag New York Inc.
eBook ISBN
978-1-4612-6374-6
DOI
10.1007/978-1-4612-6374-6
Softcover ISBN
978-0-387-90200-5
Series ISSN
0066-5452
Edition Number
1
Number of Pages
408
Topics

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