Averaging Methods in Nonlinear Dynamical Systems

By: Jan A. Sanders, Ferdinand Verhulst, James Murdock

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Hardcover | 6 July 2007 | Edition Number 2

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460 Pages

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Revised

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23.39 x 15.6 x 2.54

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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 the Averaging Methods in Nonlinear Dynamical Systems book. There are many changes, corrections and updates in chapters on Basic Material and Asymptotics, Averaging, and Attraction. Chapters on Periodic Averaging and Hyperbolicity, Classical (first level) Normal Form Theory, Nilpotent (classical) Normal Form, and Higher Level Normal Form Theory are entirely new and represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. Also new are surveys on invariant manifolds in Appendix C and averaging for PDEs in Appendix E. Since the first edition, the book has expanded in length and the third author, James Murdock has been added.

Review of First Edition

"One of the most striking features of the book is the nice 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 profuse, illuminating diagrams." - Mathematical Reviews

Industry Reviews

From the reviews of the second edition:

"This monograph is a second edition comprising a thorough revision and an expansion of the first one ... . Thus, the reader is exposed to the practice of examining both concrete applications ... and theory. ... comprehensive content is extremely well written. ... The presentation is self-contained, thus offering a high level and convenient exposition to those who wish to study the subject matter as a whole. ... The text introduces particular notations which are not too common in the literature." (Zvi Artstein, Mathematical Reviews, Issue 2008 h)

"The new book will be an ideal place to learn about averaging, including what's new in the last quarter century ... . Overall, the authors are to be commended for writing this timely and important piece of scholarship, which should teach many about their topic and related analysis." (Robert E. O' Malley, Jr., Siam Review, Vol. 51 (1), 2009)

"This monograph is a revised and expanded second edition of the original text from 1985 by the first two authors. This first edition has since then become one of the standard references for the modern theory of averaging and singular perturbations of ordinary differential equations. Without doubt the second edition will continue this legacy. ... The book is well-written with full proves and a wealth of enlightening examples. It can only be warmly recommended to everybody working in this field ... ." (G. Teschl, Monatshefte fuer Mathematik, Vol. 156 (4), April, 2009)

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

Averaging Methods in Nonlinear Dynamical Systems

At a Glance

Hardcover

Industry Reviews

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