The Hopf Bifurcation and Its Applications

1. Front Matter

   Pages i-xiii

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2. Introduction to Stability and Bifurcation in Dynamical Systems and Fluid Mechanics

   • J. E. Marsden, M. McCracken

   Pages 1-26

3. The Center Manifold Theorem

   • J. E. Marsden, M. McCracken

   Pages 27-49

4. Some Spectral Theory

   • J. E. Marsden, M. McCracken

   Pages 50-55

5. The Poincaré Map

   • J. E. Marsden, M. McCracken

   Pages 56-62

6. The Hopf Bifurcation Theorem in ℝ2 and in ℝn

   • J. E. Marsden, M. McCracken

   Pages 63-84

7. Other Bifurcation Theorems

   • J. E. Marsden, M. McCracken

   Pages 85-90

8. More General Conditions for Stability

   • J. E. Marsden, M. McCracken

   Pages 91-94

9. Hopf’s Bifurcation Theorem and the Center Theorem of Liapunov

   • Dieter S. Schmidt

   Pages 95-103

10. Computation of the Stability Condition

    • J. E. Marsden, M. McCracken

    Pages 104-130

11. How to use the Stability Formula; An Algorithm

    • J. E. Marsden, M. McCracken

    Pages 131-135

12. Examples

    • J. E. Marsden, M. McCracken

    Pages 136-150

13. Hopf Bifurcation and the Method of Averaging

    • S. Chow, J. Mallet-Paret

    Pages 151-162

14. A Translation of Hopf’s Original Paper

    • L. N. Howard, N. Kopell

    Pages 163-193

15. Editorial Comments

    • L. N. Howard, N. Kopell

    Pages 194-205

16. The Hopf Bifurcation Theorem for Diffeomorphisms

    • J. E. Marsden, M. McCracken

    Pages 206-218

17. The Canonical Form

    • J. E. Marsden, M. McCracken

    Pages 219-223

18. Bifurcations with Symmetry

    • Steve Schecter

    Pages 224-229

19. Bifurcation Theorems for Partial Differential Equations

    • J. E. Marsden, M. McCracken

    Pages 250-257

20. Notes on Nonlinear Semigroups

    • J. E. Marsden, M. McCracken

    Pages 258-284

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.

• Bifurcation
• Calculation
• Invariant
• Manifold
• Morphism
• dynamical systems
• equation
• proof
• theorem
• matrix theory

• Department of Mathematics, University of California at Berkeley, USA

  J. E. Marsden

• Department of Mathematics, University of California at Santa Cruz, USA

  M. McCracken

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