The Hopf Bifurcation and Its Applications

J. E. Marsden, M. McCracken (auth.)9780387902005, 0387902007

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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Table of contents :
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 Poincaré Map….Pages 56-62
The Hopf Bifurcation Theorem in ℝ 2 and in ℝ n ….Pages 63-84
Other Bifurcation Theorems….Pages 85-90
More General Conditions for Stability….Pages 91-94
Hopf’s Bifurcation Theorem and the Center Theorem of Liapunov….Pages 95-103
Computation of the Stability Condition….Pages 104-130
How to use the Stability Formula; An Algorithm….Pages 131-135
Examples….Pages 136-150
Hopf Bifurcation and the Method of Averaging….Pages 151-162
A Translation of Hopf’s Original Paper….Pages 163-193
Editorial Comments….Pages 194-205
The Hopf Bifurcation Theorem for Diffeomorphisms….Pages 206-218
The Canonical Form….Pages 219-223
Bifurcations with Symmetry….Pages 224-229
Bifurcation Theorems for Partial Differential Equations….Pages 250-257
Notes on Nonlinear Semigroups….Pages 258-284
Bifurcations in Fluid Dynamics and the Problem of Turbulence….Pages 285-303
On a Paper of G. Iooss….Pages 304-314
On a Paper of Kirchgässner and Kielhöffer….Pages 315-326
Bifurcation Phenomena in Population Models….Pages 327-353
A Mathematical Model of Two Cells Via Turing’s Equation….Pages 354-367
A Strange, Strange Attractor….Pages 368-381
Back Matter….Pages 382-408