Methods of Bifurcation Theory

Description

An alternative title for this book would perhaps be Nonlinear Analysis, Bifurcation Theory and Differential Equations. Our primary objective is to discuss those aspects of bifurcation theory which are particularly meaningful to differential equations. To accomplish this objective and to make the book accessible to a wider we have presented in detail much of the relevant background audience, material from nonlinear functional analysis and the qualitative theory of differential equations. Since there is no good reference for some of the mate­ rial, its inclusion seemed necessary. Two distinct aspects of bifurcation theory are discussed-static and dynamic. Static bifurcation theory is concerned with the changes that occur in the structure of the set of zeros of a function as parameters in the function are varied. If the function is a gradient, then variational techniques play an important role and can be employed effectively even for global problems. If the function is not a gradient or if more detailed information is desired, the general theory is usually local. At the same time, the theory is constructive and valid when several independent parameters appear in the function. In differential equations, the equilibrium solutions are the zeros of the vector field. Therefore, methods in static bifurcation theory are directly applicable. Front Matter....Pages i-xv Introduction and Examples....Pages 1-18 Elements of Nonlinear Analysis....Pages 19-88 Applications of the Implicit Function Theorem....Pages 89-114 Variational Method....Pages 115-167 The Linear Approximation and Bifurcation....Pages 168-214 Bifurcation with One Dimensional Null Space....Pages 215-243 Bifurcation with Higher Dimensional Null Spaces....Pages 244-283 Some Applications....Pages 284-310 Bifurcation near Equilibrium....Pages 311-348 Bifurcation of Autonomous Planar Equations....Pages 349-367 Bifurcation of Periodic Planar Equations....Pages 368-400 Normal Forms and Invariant Manifolds....Pages 401-442 Higher Order Bifurcation near Equilibrium....Pages 443-466 Perturbation of Spectra of Linear Operators....Pages 467-490 Back Matter....Pages 491-518