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Springer The Boltzmann Equation and Its Applications
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Springer-Verlag New York Inc. Elementary Stability and Bifurcation Theory
Book SynopsisI Asymptotic Solutions of Evolution Problems.- II Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension.- III Imperfection Theory and Isolated Solutions Which Perturb Bifurcation.- IV Stability of Steady Solutions of Evolution Equations in Two Dimensions and nDimensions.- V Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions.- VI Methods of Projection for General Problems of Bifurcation into Steady Solutions.- VII Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions.- VIII Bifurcation of Periodic Solutions in the General Case.- IX Subharmonic Bifurcation of Forced T-Periodic Solutions.- X Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions.- XI Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf's Type) in the Autonomous Case.- XII Stability and Bifurcation in Conservative Systems.Table of ContentsI 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 Locking.- Appendix X.1 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 Solutions Which Bifurcate at Rational Points of Higher Order Using the Method of Two Times.- Exercise.- Notes.- 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 $$\mathbb{J}$$ 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 = 1in the Semi-Simple Case.- XI.14 “Subharmonic” Bifurcation When n =1 in the Case When Zero is an Index-Two Double Eigenvalue of Jo.- 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 ?0/?0Irrational.- 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.- XII Stability and Bifurcation in Conservative Systems.- XII.1 The Rolling Ball.- 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.
£71.24
Springer New York Dynamics and Bifurcations
Book SynopsisEquations in dimensions one and two constitute the majority of the text, and in particular it is demonstrated that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations.Trade ReviewJ.K. Hale, H. Kocak, and H. Buttanri Dynamics and Bifurcations "This book takes the reader step by step through the vast subject of dynamical systems. Proceeding from 1 to 2 dimensions and onto higher dimensions in separate self-contained sections, the text is mathematically rigorous yet devoid of excess formalism. A refreshing balance is further achieved by the use of many excellent illustrations and a wealth of worked and unworked examples."—MATHEMATIKATable of ContentsI: Dimension One.- 1. Scalar Autonomous Equations.- 1.1. Existence and Uniqueness.- 1.2. Geometry of Flows.- 1.3. Stability of Equilibria.- 1.4. Equations on a Circle.- 2. Elementary Bifurcations.- 2.1. Dependence on Parameters - Examples.- 2.2. The Implicit Function Theorem.- 2.3. Local Perturbations Near Equilibria.- 2.4. An Example on a Circle.- 2.5. Computing Bifurcation Diagrams.- 2.6. Equivalence of Flows.- 3. Scalar Maps.- 3.1. Euler’s Algorithm and Maps.- 3.2. Geometry of Scalar Maps.- 3.3. Bifurcations of Monotone Maps.- 3.4. Period-doubling Bifurcation.- 3.5. An Example: The Logistic Map.- II: Dimension One and One Half.- 4. Scalar Nonautonomous Equations.- 4.1. General Properties of Solutions.- 4.2. Geometry of Periodic Equations.- 4.3. Periodic Equations on a Cylinder.- 4.4. Examples of Periodic Equations.- 4.5. Stability of Periodic Solutions.- 5. Bifurcation of Periodic Equations.- 5.1. Bifurcations of Poincaré Maps.- 5.2. Stability of Nonhyperbolic Periodic Solutions.- 5.3. Perturbations of Vector Fields.- 6. On Tori and Circles.- 6.1. Differential Equations on a Torus.- 6.2. Rotation Number.- 6.3. An Example: The Standard Circle Map.- III: Dimension Two.- 7. Planar Autonomous Systems.- 7.1. “Natural” Examples of Planar Systems.- 7.2. General Properties and Geometry.- 7.3. Product Systems.- 7.4. First Integrals and Conservative Systems.- 7.5. Examples of Elementary Bifurcations.- 8. Linear Systems.- 8.1. Properties of Solutions of Linear Systems.- 8.2. Reduction to Canonical Forms.- 8.3. Qualitative Equivalence in Linear Systems.- 8.4. Bifurcations in Linear Systems.- 8.5. Nonhomogeneous Linear Systems.- 8.6. Linear Systems with 1-periodic Coefficients.- 9. Near Equilibria.- 9.1. Asymptotic Stability from Linearization.- 9.2. Instability from Linearization.- 9.3. Liapunov Functions.- 9.4. An Invariance Principle.- 9.5. Preservation of a Saddle.- 9.6. Flow Equivalence Near Hyperbolic Equilibria.- 9.7. Saddle Connections.- 10. In the Presence of a Zero Eigenvalue.- 10.1. Stability.- 10.2. Bifurcations.- 10.3. Center Manifolds.- 11. In the Presence of Purely Imaginary Eigenvalues.- 11.1. Stability.- 11.2. Poincaré-Andronov-Hopf Bifurcation.- 11.3. Computing Bifurcation Curves.- 12. Periodic Orbits.- 12.1. Poincaré-Bendixson Theorem.- 12.2. Stability of Periodic Orbits.- 12.3. Local Bifurcations of Periodic Orbits.- 12.4. A Homoclinic Bifurcation.- 13. All Planar Things Considered.- 13.1. Structurally Stable Vector Fields.- 13.2. Dissipative Systems.- 13.3. One-parameter Generic Bifurcations.- 13.4. Bifurcations in the Presence of Symmetry.- 13.5. Local Two-parameter Bifurcations.- 14- Conservative and Gradient Systems.- 14.1. Second-order Conservative Systems.- 14.2. Bifurcations in Conservative Systems.- 14.3. Gradient Vector Fields.- 15. Planar Maps.- 15.1. Linear Maps.- 15.2. Near Fixed Points.- 15.3. Numerical Algorithms and Maps.- 15.4. Saddle Node and Period Doubling.- 15.5. Poincaré-Andronov-Hopf Bifurcation.- 15.6. Area-preserving Maps.- IV: Higher Dimensions.- 16. Dimension Two and One Half.- 16.1. Forced Van der Pol.- 16.2. Forced Duffing.- 16.3. Near a Transversal Homoclinic Point.- 16.4. Forced and Damped Duffing.- 17. Dimension Three.- 17.1. Period Doubling.- 17.2. Bifurcation to Invariant Torus.- 17.3. Silnikov Orbits.- 17.4. The Lorenz Equations.- 18. Dimension Four.- 18.1. Integrable Hamiltonians.- 18.2. A Nonintegrable Hamiltonian.- Farewell.- APPENDIX: A Catalogue of Fundamental Theorems.- References.
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