Elementary Stability and Bifurcation Theory

1. Front Matter

   Pages i-xxiii

   PDF

2. Asymptotic Solutions of Evolution Problems

   • Gérard Iooss, Daniel D. Joseph

   Pages 1-9

3. Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension

   • Gérard Iooss, Daniel D. Joseph

   Pages 10-28

4. Imperfection Theory and Isolated Solutions Which Perturb Bifurcation

   • Gérard Iooss, Daniel D. Joseph

   Pages 29-41

5. Stability of Steady Solutions of Evolution Equations in Two Dimensions and n Dimensions

   • Gérard Iooss, Daniel D. Joseph

   Pages 42-58

6. Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions

   • Gérard Iooss, Daniel D. Joseph

   Pages 59-86

7. Methods of Projection for General Problems of Bifurcation into Steady Solutions

   • Gérard Iooss, Daniel D. Joseph

   Pages 87-138

8. Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions

   • Gérard Iooss, Daniel D. Joseph

   Pages 139-155

9. Bifurcation of Periodic Solutions in the General Case

   • Gérard Iooss, Daniel D. Joseph

   Pages 156-176

10. Subharmonic Bifurcation of Forced T-Periodic Solutions

    • Gérard Iooss, Daniel D. Joseph

    Pages 177-207

11. Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions

    • Gérard Iooss, Daniel D. Joseph

    Pages 208-255

12. Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf’s Type) in the Autonomous Case

    • Gérard Iooss, Daniel D. Joseph

    Pages 256-302

13. Stability and Bifurcation in Conservative Systems

    • Gérard Iooss, Daniel D. Joseph

    Pages 303-318

14. Back Matter

    Pages 319-326

    PDF

In its most general form bifurcation theory is a theory of asymptotic solutions of nonlinear equations. By asymptotic solutions we mean, for example, steady solutions, time-periodic solutions, and quasi-periodic solutions. The purpose of this book is to teach the theory of bifurcation of asymptotic solutions of evolution problems governed by nonlinear differential equations. We have written this book for the broadest audience of potentially interested learners: engineers, biologists, chemists, physicists, mathematicians, economists, and others whose work involves understanding asymptotic solutions of nonlinear differential equations. To accomplish our aims, we have thought it necessary to make the analysis: (1) general enough to apply to the huge variety of applications which arise in science and technology; and (2) simple enough so that it can be understood by persons whose mathe­ matical training does not extend beyond the classical methods of analysis which were popular in the nineteenth century. Of course, it is not possible to achieve generality and simplicity in a perfect union but, in fact, the general theory is simpler than the detailed theory required for particular applications. The general theory abstracts from the detailed problems only the essential features and provides the student with the skeleton on which detailed structures of the applications must rest. lt is generally believed that the mathematical theory of bifurcation requires some functional analysis and some ofthe methods of topology and dynamics.

• Bifurcation
• Nichtlineare Entwicklungsgleichung
• Potential
• Stability
• Stabilität
• Verzweigung (Math.)
• differential equation

• Institut Non Lineaire, de Nice, Valbonne, France

  Gérard Iooss

• Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, USA

  Daniel D. Joseph

Policies and ethics