Dynamics reported : expositions in dynamical systems : new series

著者

• Jones, C. K. R. T. (Christopher K. R. T.)
• Kirchgraber, Urs
• Walther, Hans-Otto
• Bielawski, R.
• Fournier, G. (Gilles)
• Blokh, A. M.
• Dumas, H. S
• Fenichel, Gerald M.

書誌事項

Dynamics reported : expositions in dynamical systems : new series

C.K.R.T. Jones, U. Kirchgraber, H.O. Walther, managing editors

Springer-Verlag, c1992-

• v. 1 : gw
• v. 1 : us
• v. 2 : gw
• v. 2 : us
• v. 3 : gw
• v. 3 : us
• v. 4 : gw
• v. 4 : us
• v. 5 : gw
• v. 5 : us

注記

Vol. 1: With contributions of R. Bielawski ... [et al]

Vol. 2: With contributions of H.S. Dumas ... [et al.]

Vol. 3: With contributions of G. Fournier ... [et al.]

Vol. 4: With contributions of A.M. Blokh ... [et al.]

Vol. 5: With contributions of A.M. Fenichel ...[et al.]

Includes bibliographical references

内容説明・目次

巻冊次

v. 1 : gw ISBN 9783540541936

内容説明

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.

目次

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.

巻冊次

v. 2 : gw ISBN 9783540564096

内容説明

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.

目次

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.

巻冊次

v. 3 : gw ISBN 9783540567271

内容説明

This volume contains research papers that contribute to the field of dynamical systems. They cover limit relative category and critical point theory, the co-existence of infinitely many stable solutions to reaction diffusion systems and second-order hyperbolic mixed problems.

巻冊次

v. 4 : gw ISBN 9783540583912

内容説明

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 processes 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 subjects in dy- namical systems and their applications, addressed not only to specialists but also to a broader range of readers including graduate students. Topics are advanced, while detailed exposition of ideas, restriction to typical results - rather than the most general one- 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 entering the field and will stimulate an exchange of ideas among those working in dynamical systems Summer 1991 Christopher K. R. T Jones Drs Kirchgraber Hans-Otto Walther Managing Editors Table of Contents The "Spectral" Decomposition for One-Dimensional Maps Alexander M. Blokh Introduction and Main Results 1. 1 Preliminaries ...1 1. 0. 1. 1. Historical Remarks ...2 1. 2. A Short Description of the Approach Presented ...3 1. 3. Solenoidal Sets ...4 Basic Sets ...1. 4.

目次

The "Spectral" Decomposition for One-Dimensional Maps.- 1. Introduction and Main Results.- 1.0 Preliminaries.- 1.1 Historical Remarks.- 1.2 A Short Description of the Approach Presented.- 1.3 Solenoidal Sets.- 1.4 Basic Sets.- 1.5 The Decomposition and Main Corollaries.- 1.6 The Limit Behavior and Generic Limit Sets for Maps Without Wandering Intervals.- 1.7 Topological Properties of Sets $$\overline {Per\,f}$$, w(f) and ?(f).- 1.8 Properties of Transitive and Mixing Maps.- 1.9 Corollaries Concerning Periods of Cycles for Interval Maps.- 1.10 Invariant Measures for Interval Maps.- 1.11 The Decomposition for Piecewise-Monotone Maps.- 1.12 Properties of Piecewise-Monotone Maps of Specific Kinds.- 1.13 Further Generalizations.- 2. Technical Lemmas.- 3. Solenoidal Sets.- 4. Basic Sets.- 5. The Decomposition.- 6. Limit Behavior for Maps Without Wandering Intervals.- 7. Topological Properties of the Sets Per f, ?(f) and ?(f).- 8. Transitive and Mixing Maps.- 9. Corollaries Concerning Periods of Cycles.- 10. Invariant Measures.- 11. Discussion of Some Recent Results of Block and Coven and Xiong Jincheng.- References.- A Constructive Theory of Lagrangian Tori and Computer-assisted Applications.- 1. Introduction.- 2. Quasi-Periodic Solutions and Invariant Tori for Lagrangian Systems: Algebraic Structure.- 2.1 Setup and Definitions.- 2.2 Approximate Solutions and Newton Scheme.- 2.3 The Linearized Equation.- 2.4 Solution of the Linearized Equation.- 3. Quasi-Periodic Solutions and Invariant Tori for Lagrangian Systems: Quantitative Analysis.- 3.1 Spaces of Analytic Functions and Norms.- 3.2 Analytic Tools.- 3.3 Norm-Parameters.- 3.4 Bounds on the Solution of the Linearized Equation.- 3.5 Bounds on the New Error Term.- 4. KAM Algorithm.- 4.1. A Self-Contained Description of the KAM Algorithm.- 5. A KAM Theorem.- 6. Application of the KAM Algorithm to Problems with Parameters.- 6.1 Convergent-Power-Series (Lindstedt-Poincare-Moser Series).- 6.2 Improving the Lower Bound on the Radius of Convergence.- 7. Power Series Expansions and Estimate of the Error Term.- 7.1 Power Series Expansions.- 7.2 Truncated Series as Initial Approximations and the Majorant Method.- 7.3 Numerical Initial Approximations.- 8. Computer Assisted Methods.- 8.1 Representable Numbers and Intervals.- 8.2 Intervals on VAXes.- 8.3 Interval Operations.- 9. Applications: Three-Dimensional Phase Space Systems.- 9.1 A Forced Pendulum.- 9.2 Spin-Orbit Coupling in Celestial Mechanics.- 10. Applications: Symplectic Maps.- 10.1 Formalism.- 10.2 The Newton Scheme, the Linearized Equation, etc.- 10.3 Results.- Appendices.- References.- Ergodicity in Hamiltonian Systems.- 0. Introduction.- 1. A Model Problem.- 2. The Sinai Method.- 3. Proof of the Sinai Theorem.- 4. Sectors in a Linear Symplectic Space.- 5. The Space of Lagrangian Subspaces Contained in a Sector.- 6. Unbounded Sequences of Linear Monotone Maps.- 7. Properties of the System and the Formulation of the Results.- 8. Construction of the Neighborhood and the Coordinate System.- 9. Unstable Manifolds in the Neghborhood U.- 10. Local Ergodicity in the Smooth Case.- 11. Local Ergodicity in the Discontinous Case.- 12. Proof of Sinai Theorem.- 13. 'Tail Bound'.- 14. Applications.- References.- Linearization of Random Dynamical Systems.- 1. Introduction.- 2. Random Difference Equations.- 2.1 Preliminaries.- 2.2 Quasiboundedness and Its Consequences.- 2.3 Random Invariant Fiber Bundles.- 2.4 Asymptotic Phases.- 2.5 Topological Decoupling.- 2.6 Topological Linearization.- 3. Random Dynamical Systems.- 3.1 Preliminaries and Hypotheses.- 3.2 Random Invariant Manifolds.- 3.3 Asymptotic Phases.- 3.4 The Hartman-Grobman Theorems.- 4. Local Results.- 4.1 The Discrete-Time Case.- 4.2 The Continuous-Time Case.- 5. Appendix.- References.

巻冊次

v. 5 : gw ISBN 9783540601654

内容説明

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 processes 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 subjects in dynam- ical systems and their applications, addressed not only to specialists but also to a broader range of readers including graduate students. Topics are advanced, while detailed expo- sition of ideas, restriction to typical results - 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 entering the field and will stimulate an exchange of ideas among those working in dynamical systems Summer 1991 Christopher K. R. T Jones Drs Kirchgraber Hans-Otto Walther Managing Editors Table of Contents Hyperbolicity and Exponential Dichotomy for Dynamical Systems Neil Fenichel 1. Introduction ...I 2. The Main Lemma ...2 3. The Linearization Theorem of Hartman and Grobman 5 4. Hyperbolic Invariant Sets: -orbits and Stable Manifolds 6 5.

目次

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.

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詳細情報

• NII書誌ID(NCID): BA1413029X
• ISBN:
  • 3540541934
  • 0387541934
  • 3540564098
  • 0387564098
  • 3540567275
  • 0387567275
  • 3540583912
  • 0387583912
  • 3540601651
  • 0387601651
• LCCN: 91023213
• 出版国コード: gw
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Berlin ; Tokyo
• ページ数/冊数: v.
• 大きさ: 24 cm
• 分類:
  • LCC : QA614.8
  • DC20 : 003/.85
• 件名:
  • LCSH : Differentiable dynamical systems