Bifurcation Theory: An Introduction with Applications to PDEs

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