The goal of these notes is to give a reasonahly comยญ plete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to speยญ cific problems, including stability calculations. Historicalยญ 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 "Poincareยญ 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 Ruelleยญ 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
CONTENT
Section 1 Introduction to Stability and Bifurcation in Dynamical Systems and Fluid Dynamics -- Section 2 The Center Manifold Theorem -- Section 2A Some Spectral Theory -- Section 2B The Poincarรฉ Map -- Section 3 The Hopf Bifurcation Theorem in R2 and in Rn -- Section 3A Other Bifurcation Theorems -- Section 3B More General Conditions for Stability -- Section 3C Hopfโ{128}{153}s Bifurcation Theorem and the Center Theorem of Liapunov -- Section 4 Computation of the Stability Condition -- Section 4A How to use the Stability Formula; An Algorithm -- Section 4B Examples -- Section 4C Hopf Bifurcation and the Method of Averaging -- Section 5 A Translation of Hopfโ{128}{153}s Original Paper -- Section 5A Editorial Comments -- Section 6 The Hopf Bifurcation Theorem Diffeomorphisms -- Section 6A The Canonical Form -- Section 7 Bifurcations with Symmetry -- Section 8 Bifurcation Theorems for Partial Differential Equations -- Section 8A Notes on Nonlinear Semigroups -- Section 9 Bifurcation in Fluid Dynamics and the Problem of Turbulence -- Section 9A On a Paper of G. Iooss -- Section 9B On a Paper of Kirchgรคssner and Kielhรถffer -- Section 10 Bifurcation Phenomena in Population Models -- Section 11 A Mathematical Model of Two Cells -- Section 12 A Strange, Strange Attractor -- References
