Dynamics Reported: Expositions in Dynamical Systems

ISBN-13: 9783642647581 / Angielski / Miękka / 2011 / 250 str.

G. Vegter

Opis

Dynamics Reported is a series of books dedicated to the exposition of the mathematics of dynamcial systems. Its aim is to make the recent research accessible to advanced students and younger researchers. The series is also a medium for mathematicians to use to keep up-to-date with the work being done in neighboring fields. The style is best described as expository, but complete. Thus, there is an emphasis on examples and explanations, but also theorems normally occur with their proofs. The focus is on the analytic approach to dynamical systems, emphasizing the origins of the subject in the theory of differential equations. Dynamics Reported provides an excellent foundation for seminars on dynamical systems.

Spis treści

Bifurcational Aspects of Parametric Resonance.- 1. Setting of the Problem.- a. Introduction.- b. The Parametrically Forced Pendulum, Motivation.- c. Setting of the Problem, Outline of the Method.- 2. A Normal Form Theory.- a. Preliminaries.- b. Subharmonics: A Covering Space and Discrete Symmetry.- c. The Normal Form.- 3. Computation of a Third-Order Normal Form.- 4. The Planar Hamilton Form.- a. Introduction.- b. The Type of the Origin as a Singularity, Part of the Bifurcation Diagram.- c. Bifurcations in the Case ak?0.- d. Some Remarks on a Case of Small ak?0.- e. Nearby Higher Order Subharmonics.- 5. Dynamical Conclusions.- a. (Sub-) harmonics and Their Invariant Manifolds.- b. Quasi-Periodic Invariant Circles.- c. Miscellaneous Remarks.- References.- A Survey of Normalization Techniques Applied to Perturbed Keplerian Systems.- 1. Introduction.- 2. Perturbed Keplerian Systems.- 2.1 Basic Definitions.- 2.2 Quadratic Zeeman Effect.- 2.3 Orbiting Dust.- 2.4 A Lunar Problem.- 2.5 The Artificial Satellite.- 3. Normal Form: Theory.- 3.1 Regularization.- 3.2 Lie Series and Normal Form.- 3.3 Constrained Normal Form.- 4. Normal Form: Practice.- 4.1 Extensions to a Constrained System.- 4.2 Normal Form for the Quadratic Zeeman Effect.- 4.3 Other Examples of Constrained Normal Form.- 4.4 Normalization of the Reduced Hamiltonian.- 4.5 Normal Form and Symmetry.- 5. Reduction to One Degree of Freedom.- 5.1 Reduction of the Keplerian Symmetry.- 5.2 Reduction of the Axial Symmetry.- 6. Analysis of Normalized Hamiltonian.- 6.1 The Second Reduced Hamiltonian.- 6.2 Level Sets of Hl,c: I.- 6.3 Singular Toral Fibration: I.- 6.4 Level Sets of KL,HII.- 6.5 Singular Toral Fibration: II.- 7. Appendix 1.- 8. Appendix 2.- References.- On Littlewood’s Counterexample of Unbounded Motions in Superquadratic Potentials.- 1. Introduction.- 2. Results.- 3. Proof of the Theorem.- 3.1 An Outline of the Construction.- 3.2 Modification of the Potential.- 3.3 Proof of the Resonance Condition.- 3.4 End of the Proof of the Theorem.- 3.5 Proof of Lemma 1.- 3.6 Proof of Lemma 2.- 3.7 Appendix: An Alternative Proof of the Estimate on ?(0).- References.- Center Manifold Theory in Infinite Dimensions.- 1. Introduction.- 2. General Theory.- 2.1 Main Theorems.- 2.2 Special Cases.- 3. Spectral Theory.- 4. Examples.- 5. Application to Hydrodynamic Stability Problems.- 5.1 The Classical Navier-Stokes Equations.- 5.2 Stationary Navier-Stokes Equations in a Cylinder.- References.- Oscillations in Singularly Perturbed Delay Equations.- 1. Difference Equations.- 2. Singular Perturbations of Difference Equations with Continuous Argument: Simplest Properties.- 3. Continuous Dependence on a Paramete.- 4. Impact of Singular Perturbations: Examples.- 5. Attractors of Interval Maps and Asymptotic Behavior of Solutions.- 6. Existence of Periodic Solutions.- 7. Concluding Remarks and Open Questions.- References.- Topological Approach to Differential Inclusions on Closed Subsets of ?n.- 1. Multivalued Mappings.- 2. Topological Degree of Admissible Mappings in ?n.- 3. Aronszajn’s Result.- 4. Selectionable and ?-Selectionable Multivalued Maps.- 5. Differential Inclusions in ?n.- 6. Periodic Solutions of Differential Inclusions in ?n.- 7. Sets with Property p.- 8. Contingent Cone Valued Maps.- 9. Differential Inclusions on Sets with Property p.- 10. Proof of Lemma (7.3).- References.

O autorze

Dynamics Reported is a series of books covering recent research in the mathematics of dynamical systems. The style is expository, but at the research level. Roughly one book per year is planned. The material is ideally suited to be the foundation of seminars on dynamical systems, involving advanced students and active mathematicians.