Cesari / Hale / Lasalle Dynamical Systems

An International Symposium
1. Auflage 2014
ISBN: 978-1-4832-5969-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 336 Seiten, Web PDF

Dynamical Systems: An International Symposium, Volume 2 contains the proceedings of the International Symposium on Dynamical Systemsheld at Brown University in Providence, Rhode Island, on August 12-16, 1974. The symposium provided a forum for reviewing the theory of dynamical systems in relation to ordinary and functional differential equations, as well as the influence of this approach and the techniques of ordinary differential equations on research concerning certain types of partial differential equations and evolutionary equations in general. Comprised of six chapters, this volume first examines how the theory of isolating blocks may be applied to the Newtonian planar three-body problem. The reader is then introduced to the separatrix structure for regions attracted to solitary periodic solutions; solitary invariant sets; and singular points and separatrices. Subsequent chapters focus on the equivalence of suspensions and manifolds with cross section; a geometrical approach to classical mechanics; bifurcation theory for odd potential operators; and continuous dependence of fixed points of condensing maps. This monograph will be of interest to students and practitioners in the field of applied mathematics.

Autoren/Hrsg.

Cesari, Lamberto

Hale, Jack K.

Lasalle, Joseph P.

Inhaltsverzeichnis

1;Front Cover;1
2;Dynamical Systems: An International Symposium;4
3;Copyright Page;5
4;Table of Contents;6
5;List of Contributors;12
6;Preface;18
7;Contents of Volume 1;20
8;Chapter 1. QUALITATIVE THEORY;24
8.1;Some Qualitative Aspects of the Three-Body Flow;24
8.1.1;1. The Model;24
8.1.2;2. Isolating Blocks for the Three-Body Flow;26
8.1.3;3. Unfinished Work;28
8.1.4;REFERENCES;28
8.2;Separatrix Structure for Regions Attracted to Solitary Periodic Solutions;30
8.2.1;REFERENCES;35
8.3;Solitary Invariant Sets;36
8.3.1;REFERENCES;40
8.4;Singular Points and Separatrices;42
8.4.1;0. Introduction;42
8.4.2;1. Definition of Separatrix;43
8.4.3;2. The Structure of Component of Mn\S;45
8.4.4;3. Separatrices and Singular Points;45
8.4.5;4. Separatrix Manifolds;46
8.4.6;REFERENCES;47
8.5;Global Results by Local Averaging for Nearly Hamiltonian Systems;48
8.5.1;REFERENCES;50
8.6;Equivalence of Suspensions and Manifolds with Cross Section;52
8.6.1;1. Introduction;52
8.6.2;2. Equivalence of Suspensions;53
8.6.3;3. Manifolds with Cross Section;53
8.6.4;REFERENCES;54
8.7;Structural Stability Theorems;56
8.7.1;REFERENCES;59
8.8;Numerical Studies of an Area-Preserving Mapping;60
8.8.1;REFERENCES;62
8.9;A Geometrical Approach to Classical Mechanics;64
8.9.1;1. Introduction;64
8.9.2;2. Statement of Results;65
8.9.3;3. Construction of the Curvature Functions;67
8.9.4;REFERENCES;68
9;Chapter 2. GENERAL THEORY;70
9.1;A Solution of Ulam's Conjecture on the Existence of Invariant Measures and its Applications;70
9.1.1;1. Rényi Theorem;70
9.1.2;2. Kryloff-Bogoliuboff Theorem;71
9.1.3;3. Ulam's Conjecture;72
9.1.4;4. Frobenius–Perron Operator;73
9.1.5;5. Applications;75
9.1.6;6. Final Remarks;77
9.1.7;REFERENCES;78
9.2;Bifurcation Theory for Odd Potential Operators;80
9.2.1;1. Existence of Eigenvalues and Eigenfunctions;80
9.2.2;2. Bifurcation from an Eigenvalue;81
9.2.3;REFERENCES;84
9.3;An Existence Theorem for Solutions of Orientor Fields;86
9.3.1;Introduction;86
9.3.2;1. Approximate Solutions;87
9.3.3;2. Construction of Approximate Solutions;88
9.3.4;REFERENCES;89
9.4;Nonlinear Perturbations at Resonance;90
9.4.1;REFERENCES;94
9.5;On Continuous Dependence of Fixed Points of Condensing Maps;96
9.5.1;REFERENCES;98
9.6;Small Noise Ergodic Dynamical Systems;100
9.6.1;REFERENCES;102
10;Chapter 3. EVOLUTIONARY EQUATIONS;104
10.1;"Pointwise Degeneracy" for Delay Evolutionary Equations;104
10.1.1;I. Introduction;104
10.1.2;II. Transformation of the Problem. A Necessary and Sufficient Condition;104
10.1.3;III. Sufficient Conditions. Example;106
10.1.4;IV. A Consequence of "Pointwise Degeneracy";108
10.1.5;REFERENCES;109
10.2;On Constructing a Liapunov Functional While Defining a Linear Dynamical System;110
10.2.1;REFERENCES;113
10.3;Measurability and Continuity Conditions for Evolutionary Processes;114
10.3.1;1. Introduction;114
10.3.2;2. Results for Evolutionary Processes;114
10.3.3;3. Some Counterexamples;116
10.3.4;ACKNOWLEDGMENT;117
10.3.5;REFERENCES;117
10.4;Stabilization of Linear Evolutionary Processes;118
10.4.1;REFERENCES;121
11;Chapter 4. FUNCTIONAL DIFFERENTIAL EQUATIONS;122
11.1;Bifurcation Theory and Periodic Solutions of Some Autonomous Functional Differential Equations;122
11.1.1;REFERENCES;125
11.2;A Stability Criterion for Linear Autonomous Functional Differential Equations;126
11.2.1;REFERENCES;129
11.3;Periodic Differential Difference Equations;132
11.3.1;1. The Periodic Case;132
11.3.2;2. The Nonperiodic Case;134
11.3.3;REFERENCES;135
11.4;Point Data Problems for Functional Differential Equations;138
11.4.1;REFERENCES;144
11.5;Relations between Functional and Ordinary Differential Equations;146
11.5.1;In memoriam: Solomon Lefschetz;146
11.5.2;REFERENCES;148
11.6;Asymptotically Autonomous Neutral Functional Differential Equations with Time-Dependent Lag;150
11.6.1;REFERENCES;154
11.7;The Invariance Principle for Functional Equations;156
11.7.1;REFERENCES;159
11.8;Existence and Stability of Periodic Solutions;160
11.8.1;REFERENCES;164
11.9;Existence and Stability of Solutions on the Real Line to X{t) +ft-8 a{t—t)g{t, X(t)) dt=f(t), with General Forcing Term;166
11.9.1;REFERENCES;169
11.10;Existence and Stability for Partial Functional Differential Equations;170
11.10.1;1. Introduction;170
11.10.2;2. Existence of Solutions in the Nonlinear Case;171
11.10.3;3. The Semigroup and Infinitesimal Generator in the Autonomous Case;171
11.10.4;4. The Spectral Properties of Av in the Linear Case;172
11.10.5;5. Stability of Solutions and Examples;173
11.10.6;REFERENCES;174
11.11;Periodic Solutions to a Population Equation;176
11.11.1;REFERENCES;180
11.12;Existence and Stability of Forced Oscillation in Retarded Equations;182
11.12.1;Notation;182
11.12.2;REFERENCES;185
11.13;Exact Solutions of Some Functional Differential Equations;186
11.13.1;1. Representations of Solutions of Linear Functional Differential Equations;186
11.13.2;2. Solution of Integrodifferential Equations;187
11.13.3;3. Examples;189
11.13.4;4. Generalizations;190
11.13.5;REFERENCES;191
12;Chapter 5. TOPOLOGICAL DYNAMICAL SYSTEMS;192
12.1;Extendability of an Elementary Dynamical System to an Abstract Local Dynamical System;192
12.1.1;1. Introduction;192
12.1.2;2. No-Intersection Axiom;193
12.1.3;REFERENCES;196
12.2;Skew-Product Dynamical Systems;198
12.2.1;1. Introduction;198
12.2.2;2. Bounded Solutions of Linear Systems;200
12.2.3;REFERENCES;201
12.3;Liapunov Functions and the Comparison Principle;204
12.3.1;Introduction;204
12.3.2;1. General Definitions and Notations;205
12.3.3;2. Liapunov Mappings;206
12.3.4;3. Para-Liapunov Functions;206
12.3.5;4. Weak Stability or Controllability;207
12.3.6;REFERENCES;208
12.4;Distal Semidynamical Systems;210
12.4.1;1. Notation, Definitions, and Some Known Results;210
12.4.2;2. The Main Results;211
12.4.3;ACKNOWLEDGMENT;213
12.4.4;REFERENCES;213
12.5;Prolongations in Semidynamical Systems;214
12.5.1;Introduction;214
12.5.2;Preliminaries;214
12.5.3;REFERENCES;217
12.6;When Do Lyapunov Functions Exist on Invariant Neighborhoods?;220
12.6.1;1. Notation, Definition, and Some Known Results;220
12.6.2;2. Remarks;222
12.6.3;3. New Results;222
12.6.4;ACKNOWLEDGMENTS;224
12.6.5;REFERENCES;224
12.7;The "Simplest" Dynamical System;226
12.7.1;1. Introduction;226
12.7.2;2. Applications;226
12.7.3;3. Predicting Chaos;227
12.7.4;4. Computation Observation;228
12.7.5;REFERENCES;229
12.8;Continuous Operators That Generate Many Flows;230
12.9;Existence and Continuity of Liapunov Functions in General Systems;234
12.9.1;Introduction;234
12.9.2;1. On the Existence of Liapunov Functions;235
12.9.3;2. Semicontinuity and Continuity of Liapunov Functions;236
12.9.4;REFERENCES;239
13;Chapter 6. ORDINARY DIFFERENTIAL AND VOLTERRA EQUATIONS;240
13.1;Stability under the Perturbation by a Class of Functions;240
13.1.1;REFERENCES;245
13.2;On a General Type of Second-Order Forced Nonlinear Oscillations;246
13.2.1;REFERENCES;248
13.3;Stability of Periodic Linear Systems and the Geometry of Lie Groups;250
13.3.1;REFERENCES;254
13.4;Periodic Solutions of Holomorphic Differential Equations;256
13.5;On the Newton Method of Solving Problems of the Least Squares Type for Ordinary Differential Equations;260
13.5.1;1;260
13.5.2;2;261
13.5.3;3;262
13.5.4;REFERENCES;263
13.6;A Study on Generation of Nonuniqueness;266
13.6.1;Introduction;266
13.6.2;1. Preliminary Observations;267
13.6.3;2. Perturbation of a Differential Equation;268
13.6.4;3. Construction of a Differential Equation with Almost-Periodic Coefficients;269
13.6.5;REFERENCES;271
13.7;Relative Asymptotic Equivalence with Weight t, between Two Systems of Ordinary Differential Equations;272
13.7.1;I. Introduction;272
13.7.2;II. Preliminary Lemmas;273
13.7.3;III. Applications;276
13.7.4;REFERENCES;277
13.8;Partial Peeling;278
13.8.1;Introduction;278
13.9;Partial Peeling via D-Peeling: An Example;278
13.9.1;REFERENCES;282
13.10;Boundary Value Problems for Perturbed Differential Equations;284
13.10.1;1. Introduction;284
13.10.2;2. Results;285
13.10.3;REFERENCES;287
13.11;On Stability of Solutions of Perturbed Differential Equations;288
13.11.1;1;288
13.11.2;2;289
13.11.3;3;291
13.11.4;REFERENCES;292
13.12;Stability Theory for Nonautonomous Systems;294
13.12.1;ACKNOWLEDGMENTS;297
13.12.2;REFERENCES;297
13.13;A Nonoscillation Result for a Forced Second-Order Nonlinear Differential Equation;298
13.13.1;1. Introduction;298
13.13.2;2. A Nonoscillation Theorem;298
13.13.3;3. Proof of the Theorem;300
13.13.4;REFERENCES;301
13.14;Convexity Properties and Bounds for a Class of Linear Autonomous Mechanical Systems;302
13.14.1;Introduction;302
13.14.2;REFERENCES;305
13.15;Dynamical Systems Arising from Electrical Networks;308
13.15.1;1. Introduction;308
13.15.2;2. The Dynamics;308
13.15.3;3. Reciprocity;310
13.15.4;4. Forced Degeneracy;312
13.15.5;ACKNOWLEDGMENTS;313
13.15.6;REFERENCES;313
13.16;An Invariance Principle for Vector Liapunov Functions;314
13.16.1;REFERENCES;318
13.17;Stability of a Nonlinear Volterra Equation;320
13.17.1;1. Introduction;320
13.17.2;2. Results and Discussion;321
13.17.3;REFERENCES;324
13.18;On a Class of Volterra Integrodifferential Equations;326
13.18.1;REFERENCES;329
13.19;Existence and Continuation Properties of Solutions of a Nonlinear Volterra Integral Equation;330
13.19.1;REFERENCES;333
14;Author Index;334
15;Subject Index;336