Averaging Methods in Nonlinear Dynamical Systems (eBook)

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