Elementary stability and bifurcation

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