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Basic Material and Asymptotics 1
Introduction 1
Existence and Uniqueness 2
The Gronwall Lemma 4
Concepts of Asymptotic Approximation 5
Naive Formulation of Perturbation Problems 12
Reformulation in the Standard Form 16
The Standard Form in the Quasilinear Case 17
Averaging: the Periodic Case 21
Introduction 21
Van der Pol Equation 22
A Linear Oscillator with Frequency Modulation 24
One Degree of Freedom Hamiltonian System 25
The Necessity of Restricting the Interval of Time 26
Bounded Solutions and a Restricted Time Scale of Validity 27
Counter Example of Crude Averaging 28
Two Proofs of First-Order Periodic Averaging 30
Higher-Order Periodic Averaging and Trade-Off 37
Higher-Order Periodic Averaging 37
Estimates on Longer Time Intervals 41
Modified Van der Pol Equation 42
Periodic Orbit of the Van der Pol Equation 43
Methodology of Averaging 45
Introduction 45
Handling the Averaging Process 45
LieTheory for Matrices 46
Lie Theory for Autonomous Vector Fields 47
Lie Theory for Periodic Vector Fields 48
Solving the Averaged Equations 50
Averaging Periodic Systems with Slow Time Dependence 52
Pendulum with Slowly Varying Length 54
Unique Averaging 56
Averaging and Multiple Time Scale Methods 60
Averaging: the General Case 67
Introduction 67
Basic Lemmas; the Periodic Case 68
General Averaging 72
Linear Oscillator with Increasing Damping 75
Second-Order Averaging 77
Example of Second-Order Averaging 81
Almost-Periodic Vector Fields 82
Example 84
Attraction 89
Introduction 89
Equations with Linear Attraction 90
Examples of Regular Perturbations with Attraction 93
Two Species 93
A perturbation theorem 94
Two Species, Continued 96
Examples of Averaging with Attraction 96
Anharmonic Oscillator with Linear Damping 97
Duffing's Equation with Damping and Forcing 97
Theory of Averaging with Attraction 100
An Attractor in the Original Equation 103
Contracting Maps 104
Attracting Limit-Cycles 106
Additional Examples 107
Perturbation of the Linear Terms 108
Damping on Various Time Scales 108
Periodic Averaging and Hyperbolicity 111
Introduction 111
Coupled Duffing Equations, An Example 113
Rest Points and Periodic Solutions 116
The Regular Case 116
The Averaging Case 117
Local Conjugacy and Shadowing 119
The Regular Case 120
The Averaging Case 126
Extended Error Estimate for Solutions Approaching an Attractor 128
Conjugacy and Shadowing in a Dumbbell-Shaped Neighborhood 129
The Regular Case 130
The Averaging Case 134
Extension to Larger Compact Sets 135
Extensions and Degenerate Cases 138
Averaging over Angles 141
Introduction 141
The Case of Constant Frequencies 141
Total Resonances 146
The Case of Variable Frequencies 150
Examples 152
Einstein Pendulum 152
Nonlinear Oscillator 153
Oscillator Attached to a Flywheel 154
Secondary (Not Second Order) Averaging 156
Formal Theory 157
Slowly Varying Frequency 159
Einstein Pendulum 163
Higher Order Approximation in the Regular Case 163
Generalization of the Regular Case 166
Two-Body Problem with Variable Mass 169
Passage Through Resonance 171
Introduction 171
The Inner Expansion 172
The Outer Expansion 173
The Composite Expansion 174
Remarks on Higher-Dimensional Problems 175
Introduction 175
The Case of More Than One Angle 175
Example of Resonance Locking 176
Example of Forced Passage through Resonance 178
Inner and Outer Expansion 179
Two Examples 188
The Forced Mathematical Pendulum 188
An Oscillator Attached to a Fly-Wheel 190
From Averaging to Normal Forms 193
Classical, or First-Level, Normal Forms 193
Differential Operators Associated with a Vector Field 194
Lie Theory 196
Normal Form Styles 197
The Semisimple Case 198
The Nonsemisimple Case 199
The Transpose or Inner Product Normal Form Style 200
The sl[subscript 2] Normal Form 201
Higher Level Normal Forms 202
Hamiltonian Normal Form Theory 205
Introduction 205
The Hamiltonian Formalism 205
Local Expansions and Rescaling 207
Basic Ingredients of the Flow 207
Normalization of Hamiltonians around Equilibria 210
The Generating Function 210
Normal Form Polynomials 213
Canonical Variables at Resonance 214
Periodic Solutions and Integrals 215
Two Degrees of Freedom, General Theory 216
Introduction 216
The Linear Flow 218
Description of the w[subscript 1]: w[subscript 2]-Resonance in Normal Form 220
General Aspects of the k : l-Resonance, k [not equal] l 221
Two Degrees of Freedom, Examples 223
The 1 : 2-Resonance 223
The Symmetric 1 : 1-Resonance 227
The 1 : 3-Resonance 229
Higher-order Resonances 233
Three Degrees of Freedom, General Theory 238
Introduction 238
The Order of Resonance 239
Periodic Orbits and Integrals 241
The w[subscript 1]: w[subscript 2]: w[subscript 3]-Resonance 243
The Kernel of ad(H[superscript 0]) 243
Three Degrees of Freedom, Examples 249
The 1 : 2 : 1-Resonance 249
Integrability of the 1 : 2 : 1 Normal Form 250
The 1 : 2 : 2-Resonance 252
Integrability of the 1 : 2 : 2 Normal Form 253
The 1 : 2 : 3-Resonance 254
Integrability of the 1 : 2 : 3 Normal Form 255
The 1 : 2 : 4-Resonance 257
Integrability of the 1 : 2 : 4 Normal Form 258
Summary of Integrability of Normalized Systems 259
Genuine Second-Order Resonances 260
Classical (First-Level) Normal Form Theory 263
Introduction 263
Leibniz Algebras and Representations 264
Cohomology 267
A Matter of Style 269
Example: Nilpotent Linear Part in R[superscript 2] 272
Induced Linear Algebra 274
The Nilpotent Case 276
Nilpotent Example Revisited 278
The Nonsemisimple Case 279
The Form of the Normal Form, the Description Problem 281
Nilpotent (Classical) Normal Form 285
Introduction 285
Classical Invariant Theory 285
Transvectants 286
A Remark on Generating Functions 290
The Jacobson-Morozov Lemma 293
Description of the First Level Normal Forms 294
The N[subscript 2] Case 294
The N[subscript 3] Case 297
The N[subscript 4] Case 298
Intermezzo: How Free? 302
The N[subscript 2,2] Case 303
The N[subscript 5] Case 306
The N[subscript 2,3] Case 307
Description of the First Level Normal Forms 310
The N[subscript 2,2,2] Case 310
The N[subscript 3,3] Case 311
The N[subscript 3,4] Case 312
Concluding Remark 314
Higher-Level Normal Form Theory 315
Introduction 315
Some Standard Results 316
Abstract Formulation of Normal Form Theory 317
The Hilbert-Poincare Series of a Spectral Sequence 320
The Anharmonic Oscillator 321
Case A[superscript r]: [Characters not reproducible] Is Invertible 323
Case A[superscript r]: [Characters not reproducible] Is Not Invertible, but [Characters not reproducible] Is 323
The m-adic Approach 326
The Hamiltonian 1 : 2-Resonance 326
Averaging over Angles 328
Definition of Normal Form 329
Linear Convergence, Using the Newton Method 330
Quadratic Convergence, Using the Dynkin Formula 334
The History of the Theory of Averaging 337
Early Calculations and Ideas 337
Formal Perturbation Theory and Averaging 340
Jacobi 340
Poincare 341
Van der Pol 342
Proofs of Asymptotic Validity 343
A 4-Dimensional Example of Hopf Bifurcation 345
Introduction 345
The Model Problem 346
The Linear Equation 347
Linear Perturbation Theory 348
The Nonlinear Problem 350
Invariant Manifolds by Averaging 353
Introduction 353
Deforming a Normally Hyperbolic Manifold 354
Tori by Bogoliubov-Mitropolsky-Hale Continuation 356
The Case of Parallel Flow 357
Tori Created by Neimark-Sacker Bifurcation 360
Celestial Mechanics 363
Introduction 363
The Unperturbed Kepler Problem 364
Perturbations 365
Motion Around an 'Oblate Planet' 366
Harmonic Oscillator Formulation 367
First Order Averaging 368
A Dissipative Force: Atmospheric Drag 371
Systems with Mass Loss or Variable G 373
Two-body System with Increasing Mass 376
On Averaging Methods for Partial Differential Equations 377
Introduction 377
Averaging of Operators 378
Averaging in a Banach Space 378
Averaging a Time-Dependent Operator 379
A Time-Periodic Advection-Diffusion Problem 381
Nonlinearities, Boundary Conditions and Sources 382
Hyperbolic Operators with a Discrete Spectrum 383
Averaging Results by Buitelaar 384
Galerkin Averaging Results 386
Example: the Cubic Klein-Gordon Equation 389
Example: Wave Equation with Many Resonances 391
Example: the Keller-Kogelman Problem 392
Discussion 394
References 395
Index of Definitions & Descriptions 413
General Index 417
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Add Averaging Methods in Nonlinear Dynamical Systems, Perturbation theory and in particular normal form theory has shown strong growth during the last decades. So it is not surprising that the authors have presented an extensive revision of the first edition of Averaging Methods in Nonlinear Dynamical System, Averaging Methods in Nonlinear Dynamical Systems to the inventory that you are selling on WonderClubX
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Add Averaging Methods in Nonlinear Dynamical Systems, Perturbation theory and in particular normal form theory has shown strong growth during the last decades. So it is not surprising that the authors have presented an extensive revision of the first edition of Averaging Methods in Nonlinear Dynamical System, Averaging Methods in Nonlinear Dynamical Systems to your collection on WonderClub |