Elementary Stability and Bifurcation Theory

This substantially revised second edition teaches the bifurcation of asymptotic solutions to evolution problems governed by nonlinear differential equations. Written not just for mathematicians, it appeals to the widest audience of learners, including engineers, biologists, chemists, physicists and economists. For this reason, it uses only well-known methods of classical analysis at foundation level, while the applications and examples are specially chosen to be as varied as possible.

Contenu

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.- II Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension.- II.1 The Implicit Function Theorem.- II.2 Classification of Points on Solution Curves.- 1I.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.1 1 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.- 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.- IV Stability of Steady Solutions of Evolution Equations in Two Dimensions and nDimensions.- IV.1 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 ?n.- IV.6 Nodes, Saddles, and Foci.- IV.7 Criticality and Strict Loss of Stability.- Appendix IV.I Biorthogonality for Generalized Eigenvectors.- Appendix IV.2 Projections.- 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.- VI Methods of Projection for General Problems of Bifurcation into Steady Solutions.- VI.1 The Evolution Equation and the Spectral Problem.- VI.2 Construction of Steady Bifurcating Solutions as Power Series in the Amplitude.- VI.3 ?1 and ?1 in Projection.- VI.4 Stability of the Bifurcating Solution.- VI.5 The Extra Little Part for ?1 in Projection.- V1.6 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 ?1.- VI.10 The Method of Projection for Isolated Solutions Which Perturb Bifurcation at a Simple Eigenvalue (Imperfection Theory).- VI.1 1 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.- 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 ?1.- 6.2 Floquet Theory in ?2 and ?n.- VII.7 Equations Governing the Stability of the Periodic Solutions.- VII.8 The Factorization Theorem.- VII.9 Interpretation of the Stability Result.- Example.- 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.- 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 ?( µ).- IX.3 Biorthogonality.- IX.4 Criticality.- IX.S The Fredholm Alternative for J( µ) -?( 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 $$\mathbb{J}$$ and its Eigenvectors.- IX.9 The Adjoint Operator $${{\mathbb{J}}^{*}}$$ Biorthogonality, Strict Crossing, and the Fredholm Alternative for $$\mathbb{J}$$.- IX.10 The Amplitude ?and the Biorthogonal Decomposition of Bifurcating Subharmonic Solutions.- IX.11 The Equations Governing the Derivatives of Bifurcating Subharmonic Solutions with Respect to ?at ? =0.- IX.12 Bifurcation and Stability of T-Periodic and 2 T-Periodic Solutions.- IX.13 Bifurcation and Stability of n T-Periodic Solutions with n >2.- IX.14 Bifurcation and Stability of 3T-Periodic Solutions.- IX.15 Bifurcation of 4 T-Periodic Solutions.- IX.16 Stability of 4 T-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.- X Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions.- X.1 Decomposition of the Solution and Amplitude Equation.- Exercise.- X.2 Derivation of the Amplitude Equation.- X.3 The Normal Equations in Polar Coordinates.- X.4 The Torus and Trajectories on the Torus in the Irrational Case.- X.5 The Torus and Trajectories on the Torus When ?0T/2? Is a Rational Point of Higher Order (n?5).- X.6 The Form of the Torus in the Case n =5.- X.7 Trajectories on the Torus When n =5.- X.8 The Form of the Torus When n >5.- X.9 Trajectories on the Torus When n?5.- X.10 Asymptotically Quasi-Periodic Solutions.- X.11 Stability of the Bifurcated Torus.- X.12 Subharmonic Solutions on the Torus.- X.13 Stability of Subharmonic Solutions on the Torus.- X.14 Frequency Lock…