Dynamics Reported

Hyperbolicity and Exponential Dichotomy for Dynamical Systems.- 1. Introduction.- 2. The Main Lemma.- 3. The Linearization Theorem of Hartman and Grobman.- 4. Hyperbolic Invariant Sets: e-orbits and Stable Manifolds.- 5. Structural Stability of Anosov Diffeomorphisms.- 6. Periodic Points of Anosov Diffeomorphisms.- 7. Axiom A Diffeomorphisms: Spectral Decomposition.- 8. The In-Phase Theorem.- 9. Flows.- 10. Proof of Lemma 1.- References.- Feedback Stabilizability of Time-Periodic ParabolicEquations.- 0. Introduction.- I. Linear Periodic Evolution Equations.- 1. The Evolution Operator.- 2. The Evolution Operator in Interpolation Spaces.- 3. Periodic Problems.- 4. Exponential Stability of the Zero Solution.- 5. The Stable and Unstable Subspaces.- 6. Autonomizing the Unstable Part.- II. Controllability, Observability and Feedback Stabilizability.- 7. The Feedback Stabilizability Problem.- 8. Finite Dimensional Theory.- 9. The Standard Assumption.- 10. Controllability and Feedback Stabilizability.- 11. Observability and Feedback Stabilizability.- III. Applications to Second Order Time-Periodic Parabolic Initial-Boundary Value Problems.- 12. Evolution Equations in Interpolation and Extrapolation Spaces.- A. Semigroups in Interpolation and Extrapolation Spaces.- B. Evolution Operators in Interpolation-Extrapolation Spaces.- C. The Cauchy-Problem.- D. Identifying the Dual of the Evolution Operator.- 13. Second Order Elliptic Boundary Value Problems.- A. Strongly Uniformly Elliptic Boundary Value Problems.- B. Function Spaces with Boundary Conditions.- C. The Lp-Realization.- D. The Dirichlet Form.- 14. Second Order Parabolic Initial-Boundary Value Problems.- A. General Assumptions.- B. The Lp-Realization.- C. Affine Perturbations.- 15. The Feedback Equation.- 16. The Free System.- A. Some Notation.- B. Regularity of the Eigenfunctions.- C. The Principal Eigenvalue.- 17. Controllability.- 18. Observability.- References.- Homoclinic Bifurcations with Weakly Expanding Center.- 1. Introduction.- 2. Hypotheses, a Reduction Principle and Basic Existence Theorems.- 3. Preliminaries.- 4. Proof of the Main Results in 2.- 5. Simple Periodic Solutions.- 6. Bifurcations of Homoclinic Solutions with One-Dimensional Local Center Manifolds.- 7. Estimates Related to a Nondegenerate Hopf Bifurcation.- 8. Interaction of Homoclinic Bifurcation and Hopf Bifurcation.- 9. The Disappearance of Periodic and Aperiodic Solutions when ?2 Passes Through Turning Points.- References.- Homoclinic Orbits in a Four Dimensional Model of a Perturbed NLS Equation: A Geometric Singular Perturbation Study.- 1. Introduction.- 1.1. Summary of Numerical Results for the 2-Mode System.- 1.2. Overview.- 2. Geometric Structure and Dynamics of the Unperturbed System.- 2.1. M0 and WS(M0) ? W?(M0).- 2.2. The Dynamics on M0.- 2.3. The Unperturbed Homoclinic Orbits and Their Relationship to the Dynamics on M0 and WS(M0) ? Wu(M0).- 3. Geometric Structure and Dynamics of the Perturbed System.- 3.1. The Persistence of M0, WS(M0), and Wu(M0) under Perturbation.- 3.2. The Dynamics on M , Near Resonance.- 4. Fiber Representations of Stable and Unstable Manifolds.- 4.1. Representations of WS(M0) and Wu(M0) through Homoclinic Orbits.- 4.2. An Intuitive Introduction to Fibrations of Stable and Unstable Manifolds.- 4.3. A Second Example.- 4.4. Fibers for WS(M0) and Wu(M0) for the Two Mode Equations.- 4.5. Properties and Characteristics of the Fiber.- 4.6. Fiber Representations for Subsets of Wu(q ) and Wlocs(A ? M ).- 5. Orbits Homoclinic to q .- 5.1. Homoclinic Coordinates and the Hyperplane 2.- 5.2. The Melnikov Function for WS(A ? M ) ? Wu(q ).- 5.3. Explicit Evaluation of the Melnikov Function at I= 1.- 5.4. The Existence of Orbits Homoclinic to q .- 6. Numerical Study of Orbits Homoclinic to q .- 6.1. Numerical Algorithm and its Validation.- 6.2. The Calculation of a Typical Homoclinic Orbit.- 6.3. A Representative Homoclinic Orbit.- 6.4. Persistence of the Orbit Homoclinic to q .- 7. The Dynamical Consequences of Orbits Homoclinic to q : The Existence and Nature of Chaos.- 7.1. Construction of the Domains of the Maps.- 7.2. Construction of the Map P0 near the Origin.- 7.3. Construction of the Map Along the Homoclinic Orbits Outside a Neighborhood of the Origin.- 7.4. The Full Poincare map, P ? P0 o P1 : ?0 ? ?0.- 7.5. Verification of the Hypotheses of the Theorem for the Two-Mode Truncation.- 7.6. Some General Remarks and a Comparison with Silnikov Orbits.- 8. Conclusion.- References.