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# Analytic Theory of Global BifurcationBoris Buffoni & John Toland

 TABLE OF CONTENTS:Preface ixChapter 1. Introduction 11.1 Example: Bending an Elastic Rod I 21.2 Principle of Linearization 51.3 Global Theory 61.4 Layout 7PART 1. LINEAR AND NONLINEAR FUNCTIONAL ANALYSIS 9Chapter 2. Linear Functional Analysis 112.1 Preliminaries and Notation 112.2 Subspaces 132.3 Dual Spaces 142.4 Linear Operators 152.5 Neumann Series 162.6 Projections and Subspaces 172.7 Compact and Fredholm Operators 182.8 Notes on Sources 20Chapter 3. Calculus in Banach Spaces 213.1 Fréchet Differentiation 213.2 Higher Derivatives 273.3 Taylor's Theorem 313.4 Gradient Operators 323.5 Inverse and Implicit Function Theorems 353.6 Perturbation of a Simple Eigenvalue 383.7 Notes on Sources 40Chapter 4. Multilinear and Analytic Operators 414.1 Bounded Multilinear Operators 414.2 Faà deBruno Formula 444.3 Analytic Operators 454.4 Analytic Functions of Two Variables 524.5 Analytic Inverse and Implicit Function Theorems 534.6 Notes on Sources 57PART 2. ANALYTIC VARIETIES 59Chapter 5. Analytic Functions on Fn 615.1 Preliminaries 615.2 Weierstrass Division Theorem 645.3 Weierstrass Preparation Theorem 655.4 Riemann Extension Theorem 665.5 Notes on Sources 69Chapter 6. Polynomials 706.1 Constant Coefficients 706.2 Variable Coefficients 746.3 Notes on Sources 77Chapter 7. Analytic Varieties 787.1 F -Analytic Varieties 787.2 Weierstrass Analytic Varieties 817.3 Analytic Germs and Subspaces 867.4 Germs of C -analytic Varieties 887.5 One-dimensional Branches 957.6 Notes on Sources 99PART 3. BIFURCATION THEORY 101Chapter 8. Local Bifurcation Theory 1038.1 A Necessary Condition 1038.2 Lyapunov-Schmidt Reduction 1048.3 Crandall-Rabinowitz Transversality 1058.4 Bifurcation from a Simple Eigenvalue 1098.5 Bending an Elastic Rod II 1118.6 Bifurcation of Periodic Solutions 1128.7 Notes on Sources 113Chapter 9. Global Bifurcation Theory 1149.1 Global One-Dimensional Branches 1149.2 Global Analytic Bifurcation in Cones 1209.3 Bending an Elastic Rod III 1219.4 Notes on Sources 124PART 4. STOKES WAVES 125Chapter 10. Steady Periodic Water Waves 12710.1 Euler Equations 12710.2 One-dimensional Formulation 13110.3 Main Equation 13710.4 A Priori Bounds and Nekrasov's Equation 14010.5 Weak Solutions Are Classical 14610.6 Notes on Sources 151Chapter 11. Global Existence of Stokes Waves 15211.1 Local Bifurcation Theory 15211.2 Global Bifurcation from = 1 15411.3 Gradients, Morse Index and Bifurcation 15711.4 Notes on Sources 159Bibliography 161Index 167Return to Book DescriptionFile created: 4/21/2017 Questions and comments to: webmaster@press.princeton.eduPrinceton University Press