This book presents the main theorems in bifurcation theory in an abstract setting and shows how they can be applied to partial differential equations. It will serve as an important reference for students and researchers in mathematics, physics, and engineering.

Klappentext

In the past three decades, bifurcation theory has matured into a well-established and vibrant branch of mathematics. This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known results. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations. In addition to existence, qualitative properties such as stability and nodal structure of bifurcating solutions are treated in depth. This volume will serve as an important reference for mathematicians, physicists, and theoretically-inclined engineers working in bifurcation theory and its applications to partial differential equations.

Inhalt

Introduction Appendix I Local Theory I.1 The Implicit Function Theorem I.2 The Method of Lyapunov-Schmidt I.3 The Lyapunov-Schmidt Reduction for Potential Operators I.4 An Implicit Function Theorem for One-Dimensional Kernels: Turning Points I.5 Bifurcation with a One-Dimensional Kernel I.6 Bifurcation Formulas (stationary case) I.7 The Principle of Exchange of Stability (stationary case) I.8 Hopf Bifurcation I.9 Bifurcation Formulas for Hopf Bifurcation I.10 A Lyapunov Center Theorem I.11 Constrained Hopf Bifurcation for Hamiltonian, Reversible, and Conservative Systems I.12 The Principle of Exchange of Stability for Hopf Bifurcation I.13 Continuation of Periodic Solutions and Their Stability I.14 Period Doubling Bifurcation and Exchange of Stability I.15 Newton Polygon I.16 Degenerate Bifurcation at a Simple Eigenvalue and Stability of Bifurcating Solutions I.17 Degenerate Hopf Bifurcation and Floquet Exponents of Bifurcating Periodic Orbits I.18 The Principle of Reduced Stability for Stationary and Periodic Solutions I.19 Bifurcation with High-Dimensional Kernels, Multiparameter Bifurcation and Application of the Principle of Reduced Stability I.20 Bifurcation from Infinity I.21 Bifurcation with High-Dimensional Kernels for Potential Operators: Variational Methods I.22 Notes and Remarks to Chapter I Appendix II Global Theory II.1 The Brouwer Degree II.2 The Leray Schauder Degree II.3 Application of the Degree in Bifurcation Theory II.4 Odd Crossing Numbers II.5 A Degree for a Class of Proper Fredholm Operators and Global Bifurcation Theorems II.6 A Global Implicit Function Theorem II.7 Change of Morse Index and Local Bifurcation for Potential Operators II.8 Notes and Remarks to Chapter II Appendix III Applications III.1 The Fredholm Property of Elliptic Operators III.2 Local Bifurcation for Elliptic Problems III.3 Free Nonlinear Vibrations III.4 Hopf Bifurcation for Parabolic Problems III.5 Global Bifurcation and Continuation for Elliptic Problems III.6 Preservation of Nodal Structure on Global Branches III.7 Smoothness and Uniqueness of Global Positive Solution Branches III.8 Notes and Remarks to Chapter III