Dynamics Reported: Expositions in Dynamical Systems

ISBN-13: 9783642647550 / Angielski / Miękka / 2011 / 235 str.

H. S. Dumas; C. Genecand; J. Henrard

Opis

DYNAMICS REPORTED reports on recent developments in dynamical systems. Dynamical systems of course originated from ordinary differential equations. Today, dynamical systems cover a much larger area, including dynamical pro cesses described by functional and integral equations, by partial and stochastic differential equations, etc. Dynamical systems have involved remarkably in recent years. A wealth of new phenomena, new ideas and new techniques are proving to be of considerable interest to scientists in rather different fields. It is not surprising that thousands of publications on the theory itself and on its various applications are appearing. DYNAMICS REPORTED presents carefully written articles on major sub jects in dynamical systems and their applications, addressed not only to special ists but also to a broader range of readers including graduate students. Topics are advanced, while detailed exposition of ideas, restriction to typical result- rather than the most general ones - and, last but not least, lucid proofs help to gain the utmost degree of clarity. It is hoped, that DYNAMICS REPORTED will be useful for those enter ing the field and will stimulate an exchange of ideas among those working in dynamical systems."

Spis treści

Transversal Homoclinic Orbits near Elliptic Fixed Points of Area-preserving Diffeomorphisms of the Plane.- 1. Introduction.- 2. Elements of the Theory of Minimal States.- 3. A Priori Lipschitz Estimates for Minimal Orbits.- 4. First Perturbation: Isolation and Hyperbolicity of Minimal Periodic Orbits.- 5. Second Perturbation: Nondegeneracy of Homoclinic Orbits.- 6. Application to Mather Sets.- 7. Special Classes of Diffeomorphisms.- References.- Asymptotic Periodicity of Markov and Related Operators.- 1. Basic Notions and Results.- 2. The Reduction Procedure.- 3. Asymptotic Periodicity of Constrictive Marcov Operators.- 4. Weakly Almost Periodic Operators.- 5. Asymptotic Periodicity of Power Bounded Operators.- 6. Asymptotic Periodicity of Operators on Signed Measures.- References.- A Nekhoroshev-Like Theory of Classical Particle Channeling in Perfect Crystals.- I. Introduction.- 2. Background and Outline of Main Results.- 2.1 Sketch of the Development of Nekhoroshev Theory.- 2.2 Brief Description of the Physics of Particle Channeling in Crystals..- 2.3 Outline of Main Results.- 3. Formulation of the Channeling Problem.- 3.1 The Perfect Crystal Model.- 3.2 The Channeling Criterion.- 3.3 Transformation to Nearly Integrable Form.- 4. Construction of the Normal Forms.- 4.1 Description of Methods.- 4.2 Notation.- 4.3 Near Identity Canonical Transformations via the Lie Method.- 4.4 Statement of the Analytic Lemma.- 4.5 The Iterative Lemma.- 4.6 Technical Estimates.- 4.7 Proof of the Analytic Lemma.- 5. The Generalized Continuum Models.- 5.1 Resonant Zones and Resonant Blocks.- 5.2 Geometric Considerations.- 5.3 The Spatial Continuum Model.- 5.4 Channeling.- 6. Concluding Remarks.- References.- The Adiabatic Invariant in Classical Mechanics.- I The Classical Adiabatic Invariant Theory.- 1. Introduction.- 2. Action-Angle Variables.- 3. Perturbation Theory.- 4. The Adiabatic Invariant.- 5. Explicit Approach to Action-Angle Variables.- 6. Extension of Perturbation Theory to the Case of Unbounded Period.- II Transition Through a Critical Curce.- 1. Introduction.- 2. Neighborhood of an Homoclinic Orbit.- 3. The Autonomous Problem Close to the Equilibrium.- 4. The Autonomous Problem Close to the Homoclinic Orbit.- 5. Traverse from Apex to Apex.- 6. Probability of Capture.- 7. Time of Transit.- 8. Change in the Invariant.- III The Paradigms.- 1. Introduction.- 2. The Pendulum.- 3. The Second Fundamental Model.- 4. The Colombo’s Top.- 5. Dissipative Forces.- IV Applications.- 1. Introduction.- 2. Passage Through Resonance of a Forced Anharmonic Oscillator.- 3. Particle Motion in a Slowly Modulated Wave.- 4. The Magnetic Bottle.- 5. Orbit-Orbit Resonances in the Solar System.- 6. Spin-Orbit Resonance in the Solar System.- Appendix 1: Variational Equations.- Appendix 2: Fixing the Unstable Equilibrium and the Time Scale..- Appendix 3: Mean Value of Ri(?i, Ji, ?) 1?i?2.- Appendix 4: Mean Value of R3 (?3, J3, ?).- Appendix 5: Estimation of the Trajectory Close to the Equilibrium..- Appendix 6: Computation of the True Time of Transit.- Appendix 7: The Diffusion Parameter in Non-Symmetric Cases.- Appendix 8: Remarks on the Paper “On the Generalization of a Theorem of A. Liapounoff”, by J. Moser (Comm. P. Appl. Math. 9, 257-271, 1958).- References.- List of Contributors.