Ebook: The Hopf Bifurcation and Its Applications

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.

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Content:
Front Matter....Pages i-xiii
Introduction to Stability and Bifurcation in Dynamical Systems and Fluid Mechanics....Pages 1-26
The Center Manifold Theorem....Pages 27-49
Some Spectral Theory....Pages 50-55
The Poincar? Map....Pages 56-62
Other Bifurcation Theorems....Pages 63-84
More General Conditions for Stability....Pages 85-90
Hopf’s Bifurcation Theorem and the Center Theorem of Liapunov....Pages 91-94
Computation of the Stability Condition....Pages 95-103
How to use the Stability Formula; An Algorithm....Pages 104-130
Examples....Pages 131-135
Hopf Bifurcation and the Method of Averaging....Pages 136-150
A Translation of Hopf’s Original Paper....Pages 151-162
Editorial Comments....Pages 163-193
The Hopf Bifurcation Theorem for Diffeomorphisms....Pages 194-205
The Canonical Form....Pages 206-218
Bifurcations with Symmetry....Pages 219-223
Bifurcation Theorems for Partial Differential Equations....Pages 224-229
Notes on Nonlinear Semigroups....Pages 250-257
Bifurcations in Fluid Dynamics and the Problem of Turbulence....Pages 258-284
On a Paper of G. Iooss....Pages 285-303
On a Paper of Kirchg?ssner and Kielh?ffer....Pages 304-314
Bifurcation Phenomena in Population Models....Pages 315-326
A Mathematical Model of Two Cells Via Turing’s Equation....Pages 327-353
A Strange, Strange Attractor....Pages 354-367
Back Matter....Pages 368-381
....Pages 382-408