Author | Chow, Shui-Nee. author |
---|---|
Title | Methods of Bifurcation Theory [electronic resource] / by Shui-Nee Chow, Jack K. Hale |
Imprint | New York, NY : Springer New York, 1982 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-8159-4 |
Descript | XV, 525 p. online resource |
1 Introduction and Examples -- 1.1. Definition of Bifurcation Surface -- 1.2. Examples with One Parameter -- 1.3. The Euler-Bernoulli Rod -- 1.4. The Hopf Bifurcation -- 1.5. Some Generic Examples -- 1.6. Dynamic Bifurcation -- 2 Elements of Nonlinear Analysis -- 2.1. Calculus -- 2.2. Local Implicit Function Theorem -- 2.3. Global Implicit Function Theorem -- 2.4. Alternative Methods -- 2.5. Embedding Theorems -- 2.6. Weierstrass Preparation Theorem -- 2.7. The Malgrange Preparation Theorem -- 2.8. Newton Polygon -- 2.9. Manifolds and Transversality -- 2.10. Sardโs Theorem -- 2.11. Topological Degree, Index of a Vector Field and Fixed Point Index -- 2.12. Ljusternik-Schnirelman Theory in ?n -- 2.13. Bibliographical Notes -- 3 Applications of the Implicit Function Theorem -- 3.1. Existence of Solutions of Ordinary Differential Equations -- 3.2. Admissible Classes in Ordinary Differential Equations -- 3.3. Global Boundary Value Problems for Ordinary Differential Equations -- 3.4. Hopf Bifurcation Theorem -- 3.5. Liapunov Center Theorem -- 3.6. Saddle Point Property -- 3.7. The Hartman-Grobman Theorem -- 3.8. An Elliptic Problem -- 3.9. A Hyperbolic Problem -- 3.10. Bibliographical Notes -- 4 Variational Method -- 4.1. Introduction -- 4.2. Weak Lower Semicontinuity -- 4.3. Monotone Operators -- 4.4. Condition (C) -- 4.5. Minimax Principle in Banach Spaces -- 4.6. Mountain Pass Theorem -- 4.7. Periodic Solutions of a Semilinear Wave Equation -- 4.8. Ljusternik-Schnirelman Theory on Banach Manifolds -- 4.9. Stationary Waves -- 4.10. The Krasnoselski Theorems -- 4.11. Variational Property of Bifurcation Equation -- 4.12. Liapunov Center Theorem at Resonance -- 4.13. Bibliographical Notes -- 5 The Linear Approximation and Bifurcation -- 5.1. Introduction -- 5.2. Eigenvalues of B -- 5.3. Eigenvalues of (B, A) -- 5.4. Eigenvalues of (B, A1, ... , AN) -- 5.5. Bifurcation from a Simple Eigenvalue -- 5.6. Applications of Simple Eigenvalues -- 5.7. Bifurcation Based on the Linear Equation -- 5.8. Global Bifurcation -- 5.9. An Application.to a Delay Differential Equation -- 5.10. Bibliographical Notes -- 6 Bifurcation with One Dimensional Null Space -- 6.1. Introduction -- 6.2. Quadratic Nonlinearities -- 6.3. Applications -- 6.4. Cubic Nonlinearities -- 6.5. Applications -- 6.6. Bifurcation from Known Solutions -- 6.7. Effects of Symmetry -- 6.8. Universal Unfoldings -- 6.9. Bibliographical Notes -- 7 Bifurcation with Higher Dimensional Null Spaces -- 7.1. Introduction -- 7.2. The Quadratic Revisited -- 7.3. Quadratic Nonlinearities I -- 7.4. Quadratic Nonlinearities II -- 7.5. Cubic Nonlinearities I -- 7.6. Cubic Nonlinearities II -- 7.7. Cubic Nonlinearities III -- 7.8. Bibliographical Notes -- 8 Some Applications -- 8.1. Introduction -- 8.2. The von Kรกrmรกn Equations -- 8.3. The Linearized Problem -- 8.4. Noncritical Length -- 8.5. Critical Length -- 8.6. An Example in Chemical Reactions -- 8.7. The Duffing Equation with Harmonic Forcing -- 8.8. Bibliographical Notes -- 9 Bifurcation near Equilibrium -- 9.1. Introduction -- 9.2. Center Manifolds -- 9.3. Autonomous Case -- 9.4. Periodic Case -- 9.5. Bifurcation from a Focus -- 9.6. Bibliographical Notes -- 10 Bifurcation of Autonomous Planar Equations -- 10.1. Introduction -- 10.2. Periodic Orbit -- 10.3. Homoclinic Orbit -- 10.4. Closed Curve with a Saddle-Node -- 10.5. Remarks on Structural Stability and Bifurcation -- 10.6. Remarks on Infinite Dimensional Systems and Turbulence -- 10.7. Bibliographical Notes -- 11 Bifurcation of Periodic Planar Equations -- 11.1. Introduction -- 11.2. Periodic Orbit-Subharmonics -- 11.3. Homoclinic Orbit -- 11.4. Subharmonics and Homoclinic Points -- 11.5. Abstract Bifurcation near a Closed Curve -- 11.6. Bibliographical Notes -- 12 Normal Forms and Invariant Manifolds -- 12.1. Introduction -- 12.2. Transformation Theory and Normal Forms -- 12.3. More on Normal Forms -- 12.4. The Method of Averaging -- 12.5. Integral Manifolds and Invariant Tori -- 12.6. Bifurcation from a Periodic Orbit to a Torus -- 12.7. Bifurcation of Tori -- 12.8. Bibliographical Notes -- 13 Higher Order Bifurcation near Equilibrium -- 13.1. Introduction -- 13.2. Two Zero Roots I -- 13.3. Two Zero Roots II -- 13.4. Two Zero Roots III -- 13.5. Several Pure Imaginary Eigenvalues -- 13.6. Bibliographical Notes -- 14 Perturbation of Spectra of Linear Operators -- 14.1. Introduction -- 14.2. Continuity Properties of the Spectrum -- 14.3. Simple Eigenvalues -- 14.4. Multiple Normal Eigenvalues -- 14.5. Self-adjoint Operators -- 14.6. Bibliographical Notes