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. Language: English