Averaging Methods in Nonlinear Dynamical Systems Book

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