An alternative title for this book would perhaps be Nonlinear Analysis, Bifurcation Theory and Differential Equations. Our primary objective is to discuss those aspects of bifurcation theory which are particularly meaningful to differential equations. To accomplish this objective and to make the book accessible to a wider we have presented in detail much of the relevant background audience, material from nonlinear functional analysis and the qualitative theory of differential equations. Since there is no good reference for some of the mate rial, its inclusion seemed necessary. Two distinct aspects of bifurcation theory are discussed-static and dynamic. Static bifurcation theory is concerned with the changes that occur in the structure of the set of zeros of a function as parameters in the function are varied. If the function is a gradient, then variational techniques play an important role and can be employed effectively even for global problems. If the function is not a gradient or if more detailed information is desired, the general theory is usually local. At the same time, the theory is constructive and valid when several independent parameters appear in the function. In differential equations, the equilibrium solutions are the zeros of the vector field. Therefore, methods in static bifurcation theory are directly applicable.
Reihe
Sprache
Verlagsort
Verlagsgruppe
Illustrationen
Dateigröße
ISBN-13
978-1-4613-8159-4 (9781461381594)
DOI
10.1007/978-1-4613-8159-4
Schweitzer Klassifikation
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. Sard's 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 Kármán 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.
