Sanders / Verhulst / Murdock Averaging Methods in Nonlinear Dynamical Systems

2. Auflage 2007
ISBN: 978-0-387-48918-6
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 59, 434 Seiten

Reihe: Applied Mathematical Sciences

Perturbation theory and in particular normal form theory has shown strong growth in recent decades. This book is a drastic revision of the first edition of the averaging book. The updated chapters represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. Also new are survey appendices on invariant manifolds. One of the most striking features of the book is the collection of examples, which range from the very simple to some that are elaborate, realistic, and of considerable practical importance. Most of them are presented in careful detail and are illustrated with illuminating diagrams.

Autoren/Hrsg.

Sanders, Jan A.

Verhulst, Ferdinand

Murdock, James

Inhaltsverzeichnis

1;Preface;6
1.1;Preface to the Revised 2nd Edition;6
1.2;Preface to the First Edition;7
2;List of Figures;9
3;List of Tables;11
4;List of Algorithms;12
5;Contents;13
6;Map of the book;20
7;1 Basic Material and Asymptotics;21
7.1;1.1 Introduction;21
7.2;1.2 The Initial Value Problem: Existence, Uniqueness and Continuation;22
7.3;1.3 The Gronwall Lemma;24
7.4;1.4 Concepts of Asymptotic Approximation;25
7.5;1.5 Naive Formulation of Perturbation Problems;32
7.6;1.6 Reformulation in the Standard Form;36
7.7;1.7 The Standard Form in the Quasilinear Case;37
8;2 Averaging: the Periodic Case;40
8.1;2.1 Introduction;40
8.2;2.2 Van der Pol Equation;41
8.3;2.3 A Linear Oscillator with Frequency Modulation;43
8.4;2.4 One Degree of Freedom Hamiltonian System;44
8.5;2.5 The Necessity of Restricting the Interval of Time;45
8.6;2.6 Bounded Solutions and a Restricted Time Scale of Validity;46
8.7;2.7 Counter Example of Crude Averaging;47
8.8;2.8 Two Proofs of First-Order Periodic Averaging;49
8.9;2.9 Higher-Order Periodic Averaging and Trade-Off;56
9;3 Methodology of Averaging;64
9.1;3.1 Introduction;64
9.2;3.2 Handling the Averaging Process;64
9.3;3.3 Averaging Periodic Systems with Slow Time Dependence;71
9.4;3.4 Unique Averaging;75
9.5;3.5 Averaging and Multiple Time Scale Methods;79
10;4 Averaging: the General Case;85
10.1;4.1 Introduction;85
10.2;4.2 Basic Lemmas; the Periodic Case;86
10.3;4.3 General Averaging;90
10.4;4.4 Linear Oscillator with Increasing Damping;93
10.5;4.5 Second-Order Approximations in General Averaging; Improved First- Order Estimate Assuming Differentiability;95
10.6;4.6 Application of General Averaging to Almost- Periodic Vector Fields;100
11;5 Attraction;106
11.1;5.1 Introduction;106
11.2;5.2 Equations with Linear Attraction;107
11.3;5.3 Examples of Regular Perturbations with Attraction;110
11.4;5.4 Examples of Averaging with Attraction;113
11.5;5.5 Theory of Averaging with Attraction;117
11.6;y;118
11.7;( t);118
11.8;x(0) y(t);118
11.9;y;118
11.10;( t);118
11.11;y;118
11.12;( t) x( t);118
11.13;5.6 An Attractor in the Original Equation;120
11.14;5.7 Contracting Maps;121
11.15;5.8 Attracting Limit-Cycles;123
11.16;5.9 Additional Examples;124
12;6 Periodic Averaging and Hyperbolicity;128
12.1;6.1 Introduction;128
12.2;6.2 Coupled Duffing Equations, An Example;130
12.3;6.3 Rest Points and Periodic Solutions;133
12.4;6.4 Local Conjugacy and Shadowing;136
12.5;6.5 Extended Error Estimate for Solutions Approaching an Attractor;145
12.6;6.6 Conjugacy and Shadowing in a Dumbbell-Shaped Neighborhood;146
12.7;6.7 Extension to Larger Compact Sets;152
12.8;6.8 Extensions and Degenerate Cases;155
13;7 Averaging over Angles;158
13.1;7.1 Introduction;158
13.2;7.2 The Case of Constant Frequencies;158
13.3;7.3 Total Resonances;163
13.4;7.4 The Case of Variable Frequencies;167
13.5;7.5 Examples;169
13.6;7.6 Secondary (Not Second Order) Averaging;173
13.7;7.7 Formal Theory;174
13.8;7.8 Systems with Slowly Varying Frequency in the Regular Case; the Einstein Pendulum;176
13.9;7.9 Higher Order Approximation in the Regular Case;180
13.10;7.10 Generalization of the Regular Case; an Example from Celestial Mechanics;183
14;8 Passage Through Resonance;188
14.1;8.1 Introduction;188
14.2;8.2 The Inner Expansion;189
14.3;8.3 The Outer Expansion;190
14.4;8.4 The Composite Expansion;191
14.5;8.5 Remarks on Higher-Dimensional Problems;192
14.6;8.6 Analysis of the Inner and Outer Expansion; Passage through Resonance;196
14.7;8.7 Two Examples;205
15;9 From Averaging to Normal Forms;210
15.1;9.1 Classical, or First-Level, Normal Forms;210
15.2;9.2 Higher Level Normal Forms;219
16;10 Hamiltonian Normal Form Theory;222
16.1;10.1 Introduction;222
16.2;10.2 Normalization of Hamiltonians around Equilibria;227
16.3;10.3 Canonical Variables at Resonance;231
16.4;10.4 Periodic Solutions and Integrals;232
16.5;10.5 Two Degrees of Freedom, General Theory;233
16.6;10.6 Two Degrees of Freedom, Examples;240
16.7;10.7 Three Degrees of Freedom, General Theory;255
16.8;10.8 Three Degrees of Freedom, Examples;266
17;11 Classical (First–Level) Normal Form Theory;280
17.1;11.1 Introduction;280
17.2;11.2 Leibniz Algebras and Representations;281
17.3;11.3 Cohomology;284
17.4;11.4 A Matter of Style;286
17.5;11.5 Induced Linear Algebra;291
17.6;11.6 The Form of the Normal Form, the Description Problem;298
18;12 Nilpotent (Classical) Normal Form;301
18.1;12.1 Introduction;301
18.2;12.2 Classical Invariant Theory;301
18.3;12.3 Transvectants;302
18.4;12.4 A Remark on Generating Functions;306
18.5;12.5 The Jacobson–Morozov Lemma;309
18.6;12.6 A GL;310
18.7;-Invariant Description of the First Level;310
18.8;Normal Forms for n < 6;310
18.9;12.7 A GL;326
18.10;-Invariant Description of the Ring of;326
18.11;Seminvariants for n 6;326
19;13 Higher–Level Normal Form Theory;331
19.1;13.1 Introduction;331
19.2;13.2 Abstract Formulation of Normal Form Theory;333
19.3;13.3 The Hilbert–Poincar e Series of a Spectral Sequence;336
19.4;13.4 The Anharmonic Oscillator;337
19.5;13.5 The Hamiltonian 1 : 2-Resonance;342
19.6;13.6 Averaging over Angles;344
19.7;13.7 Definition of Normal Form;345
19.8;13.8 Linear Convergence, Using the Newton Method;346
19.9;13.9 Quadratic Convergence, Using the Dynkin Formula;350
20;A The History of the Theory of Averaging;352
20.1;A.1 Early Calculations and Ideas;352
20.2;A.2 Formal Perturbation Theory and Averaging;355
20.3;A.3 Proofs of Asymptotic Validity;358
21;B A 4-Dimensional Example of Hopf Bifurcation;359
21.1;B.1 Introduction;359
21.2;B.2 The Model Problem;360
21.3;B.3 The Linear Equation;361
21.4;B.4 Linear Perturbation Theory;362
21.5;B.5 The Nonlinear Problem and the Averaged Equations;364
22;C Invariant Manifolds by Averaging;367
22.1;C.1 Introduction;367
22.2;C.2 Deforming a Normally Hyperbolic Manifold;368
22.3;C.3 Tori by Bogoliubov-Mitropolsky-Hale Continuation;370
22.4;C.4 The Case of Parallel Flow;371
22.5;C.5 Tori Created by Neimark–Sacker Bifurcation;374
23;D Some Elementary Exercises in Celestial Mechanics;377
23.1;D.1 Introduction;377
23.2;D.2 The Unperturbed Kepler Problem;378
23.3;D.3 Perturbations;379
23.4;D.4 Motion Around an ‘Oblate Planet’;380
23.5;D.5 Harmonic Oscillator Formulation for Motion Around an ‘ Oblate Planet’;381
23.6;D.6 First Order Averaging for Motion Around an ‘ Oblate Planet’;382
23.7;D.7 A Dissipative Force: Atmospheric Drag;385
23.8;D.8 Systems with Mass Loss or Variable G;387
23.9;D.9 Two-body System with Increasing Mass;390
24;E On Averaging Methods for Partial Differential Equations;391
24.1;E.1 Introduction;391
24.2;E.2 Averaging of Operators;392
24.3;E.3 Hyperbolic Operators with a Discrete Spectrum;397
24.4;E.4 Discussion;408
25;References;409
26;Index of Definitions & Descriptions;426
27;General Index;430