Bifurcation Theory: An Introduction with Applications to Partial Differential Equations

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