Bifurcation Theory and Applications

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World Scientific, 2005 - Science - 375 pages
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This book covers comprehensive bifurcation theory and its applications to dynamical systems and partial differential equations (PDEs) from science and engineering, including in particular PDEs from physics, chemistry, biology, and hydrodynamics.The book first introduces bifurcation theories recently developed by the authors, on steady state bifurcation for a class of nonlinear problems with even order nondegenerate nonlinearities, regardless of the multiplicity of the eigenvalues, and on attractor bifurcations for nonlinear evolution equations, a new notion of bifurcation.With this new notion of bifurcation, many longstanding bifurcation problems in science and engineering are becoming accessible, and are treated in the second part of the book. In particular, applications are covered for a variety of PDEs from science and engineering, including the Kuramoto-Sivashinsky equation, the Cahn-Hillard equation, the Ginzburg-Landau equation, reaction-diffusion equations in biology and chemistry, the Benard convection problem, and the Taylor problem. The applications provide, on the one hand, general recipes for other applications of the theory addressed in this book, and on the other, full classifications of the bifurcated attractor and the global attractor as the control parameters cross certain critical values, dictated usually by the eigenvalues of the linearized problems. It is expected that the book will greatly advance the study of nonlinear dynamics for many problems in science and engineering.
 

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Contents

Introduction to Steady State Bifurcation Theory
1
Introduction to Dynamic Bifurcation
21
Reduction Procedures and Stability
41
Steady State Bifurcations
75
Finite Dimensional Case
105
Infinite Dimensional Case
151
Bifurcations for Nonlinear Elliptic Equations
197
ReactionDiffusion Equations
241
Pattern Formation and Wave Equations
267
Fluid Dynamics
327
Bibliography
367
Index
373

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