Elementary stability and bifurcation

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Cover Title page List of Frequently Used Symbols Introduction Preface to the Second Edition CHAPTER I Asymptotic Solutions of Evolution Problems I.1 One- Dimensional, Two- Dimensional, n- Dimensional, and Infinite- Dimensional Interpretations of (I.1) I.2 Forced Solutions; Steady Forcing and T-Periodic Forcing; Autonomous and Nonautonomous Problems I.3 Reduction to Local Form I.4 Asymptotic Solutions I.5 Asymptotic Solutions and Bifurcating Solutions I.6 Bifurcating Solutions and the Linear Theory of Stability I.7 Notation for the Functional Expansion of F(t,μ,U) Notes CHAPTER II Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension II.I The Implicit Function Theorem II.2 Classification of Points on Solution Curves II.3 The Characteristic Quadratic. Double Points, Cusp Points, and Conjugate Points II.4 Double-Point Bifurcation and the Implicit Function Theorem II.5 Cusp-Point Bifurcation II.6 Triple-Point Bifurcation II.7 Conditional Stability Theorem II.8 The Factorization Theorem in One Dimension II.9 Equivalence of Strict Loss of Stability and Double-Point Bifurcation II.10 Exchange of Stability at a Double Point II.11 Exchange of Stability at a Double Point for Problems Reduced to Local Form II.12 Exchange of Stability at a Cusp Point II.13 Exchange of Stability at a Triple Point II.14 Global Properties of Stability of Isolated Solutions CHAPTER III Imperfection Theory and Isolated Solutions Which Perturb Bifurcation III.1 The Structure of Problems Which Break Double-Point Bifurcation III.2 The Implicit Function Theorem and the Saddle Surface Breaking Bifurcation III.3 Examples of Isolated Solutions Which Break Bifurcation III.4 Iterative Procedures for Finding Solutions III.5 Stability of Solutions Which Break Bifurcation III.6 Isolas Exercise Notes CHAPTER IV Stability of Steady Solutions of Evolution Equations in Two Dimensions and n Dimensions IV.l Eigenvalues and Eigenvectors of an n x n Matrix IV.2 Algebraic and Geometric Multiplicity - The Riesz Index IV.3 The Adjoint Eigenvalue Problem IV.4 Eigenvalues and Eigenvectors of a 2 x 2 Matrix 4.1 Eigenvalues 4.2 Eigenvectors 4.3 Algebraically Simple Eigenvalues 4.4 Algebraically Double Eigenvalues 4.4.1 Riesz Index 1 4.4.2 Riesz Index 2 IV.5 The Spectral Problem and Stability of the Solution u = 0 in R^n IV.6 Nodes, Saddles, and Foci IV.7 Criticality and Strict Loss of Stability Appendix IV.l Biorthogonality for Generalized Eigenvectors Appendix IV.2 Projections CHAPTER V Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions V.1 The Form of Steady Bifurcating Solutions and Their Stability V.2 Necessary Conditions for the Bifurcation of Steady Solutions V.3 Bifurcation at a Simple Eigenvalue V.4 Stability of the Steady Solution Bifurcating at a Simple Eigenvalue V.5 Bifurcation at a Double Eigenvalue of Index Two V.6 Stability of the Steady Solution Bifurcating at a Double Eigenvalue of Index Two V.7 Bifurcation and Stability of Steady Solutions in the Form (V.2) at a Double Eigenvalue of Index One (Semi-Simple) V.8 Bifurcation and Stability of Steady Solutions (V.3) at a Semi-Simple Double Eigenvalue V.9 Examples of Stability Analysis at a Double Semi-Simple (Index-One) Eigenvalue V.10 Saddle-Node Bifurcation Appendix V.1 Implicit Function Theorem for a System of Two Equations in Two Unknown Functions of One Variable Exercises CHAPTER VI Methods of Projection for General Problems of Bifurcation into Steady Solutions VI.l The Evolution Equation and the Spectral Problem VI.2 Construction of Steady Bifurcating Solutions as Power Series in the Amplitude VI.3 R¹ and R¹ in Projection VI.4 Stability of the Bifurcating Solution VI.5 The Extra Little Part for R¹ in Projection VI.6 Projections of Higher-Dimensional Problems VI.7 The Spectral Problem for the Stability of u = 0 VI.8 The Spectral Problem and the Laplace Transform VI.9 Projections into R^l VI.10 The Method of Projection for Isolated Solutions Which Perturb Bifurcation at a Simple Eigenvalue (Imperfection Theory) VI.11 The Method of Projection at a Double Eigenvalue of Index Two VI.12 The Method of Projection at a Double Semi-Simple Eigenvalue VI.13 Examples of the Method of Projection VI.14 Symmetry and Pitchfork Bifurcation CHAPTER VII Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions VII.1 The Structure of the Two-Dimensional Problem Governing Hopf Bifurcation VII.2 Amplitude Equation for Hopf Bifurcation VII.3 Series Solution VII.4 Equations Governing the Taylor Coefficients VII.5 Solvability Conditions (the Fredholm Alternative) VII.6 Floquet Theory 6.1 Floquet Theory in R¹ 6.2 Floquet Theory in R² and R^n VII.7 Equations Governing the Stability of the Periodic Solutions VII.8 The Factorization Theorem VII.9 Interpretation of the Stability Result Example CHAPTER VIII Bifurcation of Periodic Solutions in the General Case VIII.1 Eigenprojections of the Spectral Problem VIII.2 Equations Governing the Projection and the Complementary Projection VIII.3 The Series Solution Using the Fredholm Alternative VIII.4 Stability of the Hopf Bifurcation in the General Case VIII.5 Systems with Rotational Symmetry Examples Notes CHAPTER IX Subharmonic Bifurcation of Forced T-Periodic Solutions Notation IX.1 Definition of the Problem of Subharmonic Bifurcation IX.2 Spectral Problems and the Eigenvalues σ(μ) IX.3 Biorthogonality IX.4 Criticality IX.5 The Fredholm Alternative for J(μ) - σ(μ) and a Formula Expressing the Strict Crossing (IX.20) IX.6 Spectral Assumptions IX.7 Rational and Irrational Points of the Frequency Ratio at Criticality IX.8 The Operator JJ and its Eigenvectors IX.9 The Adjoint Operator JJ*, Biorthogonality, Strict Crossing, and the Fredholm Alternative for JJ IX.10 The Amplitude ε and the Biorthogonal Decomposition of Bifurcating Subharmonic Solutions IX.ll The Equations Governing the Derivatives of Bifurcating Subharmonic Solutions with Respect to ε at ε = 0 IX.12 Bifurcation and Stability of T-Periodic and 2T-Periodic Solutions IX.13 Bifurcation and Stability of nT-Periodic Solutions with n > 2 IX.14 Bifurcation and Stability of 3T-Periodic Solutions IX.15 Bifurcation of 4T-Periodic Solutions IX.16 Stability of 4T-Periodic Solutions IX.17 Nonexistence of Higher-Order Subharmonic Solutions and Weak Resonance IX.18 Summary of Results About Subharmonic Bifurcation IX.19 Imperfection Theory with a Periodic Imperfection Exercises IX.20 Saddle-Node Bifurcation of T-Periodic Solutions IX.21 General Remarks About Subharmonic Bifurcations CHAPTER X Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions X.l Decomposition of the Solution and Amplitude Equation X.2 Exercise X.3 Derivation of the Amplitude Equation X.4 The Normal Equations in Polar Coordinates X.5 The Torus and Trajectories on the Torus in the Irrational Case X.6 The Torus and Trajectories on the Torus When ω₀T/2π Is a Rational Point of Higher Order (n >= 5) X.7 The Form of the Torus in the Case n = 5 X.8 Trajectories on the Torus When n = 5 X.9 The Form of the Torus When n > 5 X.10 Trajectories on the Torus When n >= 5 X.ll Asymptotical1y Quasi- Periodic Solutions X.12 Stability of the Bifurcated Torus X.13 Subharmonic Solutions on the Torus X.14 Stability of Subharmonic Solutions on the Torus Frequency Locking Appendix X.l Direct Computation of Asymptotically Quasi-Periodic Solutions Which Bifurcate at Irrational Points Using the Method of Two Times, Power Series, and the Fredholm Alternative Appendix X.2 Direct Computation of Asymptotically Quasi-Periodic So]lutions Which Bifurcate at Rational Points of Higher Order Using the Method of Two Times Exercise Notes CHAPTER XI Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf's Type) in the Autonomous Case Notation XI.1 Spectral Problems XI.2 Criticality and Rational Points XI.3 Spectral Assumptions About J₀ XI.4 Spectral Assumptions About JJ in the Rational Case XI.5 Strict Loss of Stability at a Simple Eigenvalue of J₀ XI.6 Strict Loss of Stability at a Double Semi-Simple Eigenvalue of J₀ XI.7 Strict Loss of Stability at a Double Eigenvalue of Index Two XI.8 Formulation of the Problem of Subharmonic Bifurcation of Periodic Solutions of Autonomous Problems XI.9 The Amplitude of the Bifurcating Solution XI.10 Power-Series Solutions of the Bifurcation Problem XI.11 Subharmonic Bifurcation When n = 2 XI.12 Subharmonic Bifurcation When n > 2 XI.13 Subharmonic Bifurcation When n = 1 in the Semi-Simple Case XI.14 "Subharmonic" Bifurcation When n = 1 in the Case When Zero is an Index-Two Double Eigenvalue of J₀ XI.15 Stability of Subharmonic Solutions XI.16 Summary of Results About Subharmonic Bifurcation in the Autonomous Case XI.17 Amplitude Equations XI.18 Amplitude Equations for the Cases n >= 3 or η₀/ω₀ Irrational XI.19 Bifurcating Tori. Asymptotically Quasi-Periodic Solutions XI.20 Period Doubling, n = 2 XI.21 Pitchfork Bifurcation of Periodic Orbits in the Presence of Symmetry, n = 1 Exercises XI.22 Rotationally Symmetric Problems Exercise CHAPTER XII Stability and Bifurcation in Conservative Systems XII.1 The Rolling Ball Steady Rigid Rotation of Two Fluids XII.2 Euler Buckling Exercises XII.3 Some Remarks About Spectral Problems for Conservative Systems XII.4 Stability and Bifurcation of Rigid Rotation of Two Immiscible Liquids Steady Rigid Rotation of Two Fluids Index