Averaging Methods in Nonlinear Dynamical Systems

Front Cover
Springer Science & Business Media, Aug 18, 2007 - Mathematics - 434 pages
0 Reviews

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

 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Basic Material and Asymptotics
1
Existence Uniqueness and Continuation
2
13 The Gronwall Lemma
4
14 Concepts of Asymptotic Approximation
5
15 Naive Formulation of Perturbation Problems
12
16 Reformulation in the Standard Form
16
17 The Standard Form in the Quasilinear Case
17
Averaging the Periodic Case
20
1052 The Linear Flow
218
ω2Resonance in Normal Form
220
lResonance k l
221
106 Two Degrees of Freedom Examples
223
1Resonance
227
3Resonance
229
1064 Higherorder Resonances
233
107 Three Degrees of Freedom General Theory
238

22 Van der Pol Equation
22
23 A Linear Oscillator with Frequency Modulation
24
24 One Degree of Freedom Hamiltonian System
25
25 The Necessity of Restricting the Interval of Time
26
26 Bounded Solutions and a Restricted Time Scale of Validity
27
27 Counter Example of Crude Averaging
28
28 Two Proofs of FirstOrder Periodic Averaging
30
29 HigherOrder Periodic Averaging and TradeOff
37
292 Estimates on Longer Time Intervals
41
293 Modified Van der Pol Equation Consider the Modified Van der Pol equation
42
294 Periodic Orbit of the Van der Pol Equation
43
Methodology of Averaging
45
321 Lie Theory for Matrices
46
322 Lie Theory for Autonomous Vector Fields
47
323 Lie Theory for Periodic Vector Fields
48
324 Solving the Averaged Equations
50
33 Averaging Periodic Systems with Slow Time Dependence
52
331 Pendulum with Slowly Varying Length
54
34 Unique Averaging
56
35 Averaging and Multiple Time Scale Methods
60
Averaging the General Case
67
42 Basic Lemmas the Periodic Case
68
43 General Averaging
72
44 Linear Oscillator with Increasing Damping
75
45 SecondOrder Approximations in General Averaging Improved FirstOrder Estimate Assuming Differentiability
77
451 Example of SecondOrder Averaging
81
46 Application of General Averaging to AlmostPeriodic Vector Fields
82
461 Example
84
Attraction
88
52 Equations with Linear Attraction
90
53 Examples of Regular Perturbations with Attraction
93
532 A perturbation theorem
94
533 Two Species Continued
96
541 Anharmonic Oscillator with Linear Damping
97
55 Theory of Averaging with Attraction
100
56 An Attractor in the Original Equation
103
57 Contracting Maps
104
58 Attracting LimitCycles
106
59 Additional Examples
107
591 Perturbation of the Linear Terms
108
Periodic Averaging and Hyperbolicity 61 Introduction
111
62 Coupled Duffing Equations An Example
113
63 Rest Points and Periodic Solutions
116
632 The Averaging Case
117
64 Local Conjugacy and Shadowing
119
641 The Regular Case
120
642 The Averaging Case
126
65 Extended Error Estimate for Solutions Approaching an Attractor
128
66 Conjugacy and Shadowing in a DumbbellShaped Neighborhood
129
661 The Regular Case
130
662 The Averaging Case
134
67 Extension to Larger Compact Sets
135
68 Extensions and Degenerate Cases
138
Averaging over Angles
141
73 Total Resonances
146
74 The Case of Variable Frequencies
150
75 Examples
152
752 Nonlinear Oscillator
153
753 Oscillator Attached to a Flywheel
154
76 Secondary Not Second Order Averaging
156
77 Formal Theory
157
78 Systems with Slowly Varying Frequency in the Regular Case the Einstein Pendulum
159
781 Einstein Pendulum
163
710 Generalization of the Regular Case an Example from Celestial Mechanics
166
7101 TwoBody Problem with Variable Mass
169
Passage Through Resonance
171
82 The Inner Expansion
172
83 The Outer Expansion
173
84 The Composite Expansion
174
853 Example of Resonance Locking
176
854 Example of Forced Passage through Resonance
178
86 Analysis of the Inner and Outer Expansion Passage through Resonance
179
872 An Oscillator Attached to a FlyWheel
190
From Averaging to Normal Forms 91 Classical or FirstLevel Normal Forms
193
911 Differential Operators Associated with a Vector Field
194
912 Lie Theory
196
913 Normal Form Styles
197
914 The Semisimple Case
198
915 The Nonsemisimple Case
199
916 The Transpose or Inner Product Normal Form Style
200
917 The sl₂ Normal Form
201
92 Higher Level Normal Forms
202
Hamiltonian Normal Form Theory
205
1012 Local Expansions and Rescaling
207
102 Normalization of Hamiltonians around Equilibria
210
1022 Normal Form Polynomials
213
103 Canonical Variables at Resonance
214
104 Periodic Solutions and Integrals
215
105 Two Degrees of Freedom General Theory
216
1072 The Order of Resonance
239
1073 Periodic Orbits and Integrals
241
ω2 ω3Resonance
243
108 Three Degrees of Freedom Examples
249
21 Normal Form
250
2 2Resonance
252
22 Normal Form
253
2 3Resonance
254
23 Normal Form
255
2 4Resonance
257
24 Normal Form
258
1089 Summary of Integrability of Normalized Systems
259
10810 Genuine SecondOrder Resonances
260
Classical FirstLevel Normal Form Theory
263
112 Leibniz Algebras and Representations
264
113 Cohomology
267
114 A Matter of Style
269
Nilpotent Linear Part in K2
272
115 Induced Linear Algebra
274
1151 The Nilpotent Case
276
1152 Nilpotent Example Revisited
278
1153 The Nonsemisimple Case
279
116 The Form of the Normal Form the Description Problem
281
Nilpotent Classical Normal Form
285
123 Transvectants
286
124 A Remark on Generating Functions
290
125 The JacobsonMorozov Lemma
293
126 A GLnInvariant Description of the First Level Normal Forms for n 6
294
1262 The N3 Case
297
1263 The N4 Case
298
How Free?
302
1265 The N22 Case
303
1266 The 7V5 Case
306
1267 The N23 Case
307
127 A GLInvariant Description of the Ring of S em invariants form
310
1272 The 7V33 Case
311
1273 The N34 Case
312
1274 Concluding Remark
314
HigherLevel Normal Form Theory
315
1311 Some Standard Results
316
132 Abstract Formulation of Normal Form Theory
317
133 The HilbertPoincare Series of a Spectral Sequence
320
134 The Anharmonic Oscillator
321
3r Is Invertible
323
1343 The madic Approach
326
136 Averaging over Angles
328
137 Definition of Normal Form
329
138 Linear Convergence Using the Newton Method
330
139 Quadratic Convergence Using the Dynkin Formula
334
The History of the Theory of Averaging
336
A2 Formal Perturbation Theory and Averaging
340
A22 Poincare
341
A23 Van der Pol
342
A3 Proofs of Asymptotic Validity
343
A 4Dimensional Example of Hopf Bifurcation
345
B2 The Model Problem
346
B3 Liner Equation
347
B4 Linear Perturbation Theory
348
B5 The Nonlinear Problem and the Averaged Equations
350
Invariant Manifolds by Averaging
353
C2 Deforming a Normally Hyperbolic Manifold
354
C3 Tori by BogoliubovMitropolskyHale Continuation
356
C4 The Case of Parallel Flow
357
C5 Tori Created by NeimarkSacker Bifurcation
360
Some Elementary Exercises in Celestial Mechanics
363
D2 The Unperturbed Kepler Problem
364
D3 Perturbations
365
D4 Motion Around an Oblate Planet
366
D5 Harmonic Oscillator Formulation for Motion Around an Oblate Planet
367
D6 First Order Averaging for Motion Around an Oblate Planet
368
Atmospheric Drag
371
D8 Systems with Mass Loss or Variable G
373
D9 Twobody System with Increasing Mass
376
On Averaging Methods for Partial Differential Equations
377
E2 Averaging of Operators
378
E22 Averaging a TimeDependent Operator
379
E23 Application to a TimePer iodic Advect ionDiffusion Problem
381
E24 Nonlinearities Boundary Conditions and Sources
382
E3 Hyperbolic Operators with a Discrete Spectrum
383
E31 Averaging Results by Buitelaar
384
E32 Galerkin Averaging Results
386
the Cubic KleinGordon Equation
389
a Nonlinear Wave Equation with Infinitely Many Resonances
391
the KellerKogelman Problem
392
E4 Discussion
394
References
395
Index of Definitions Descriptions
413
General Index
417
Copyright

Other editions - View all

Common terms and phrases

Popular passages

Page ii - Courant/Friedrichs: Supersonic Flow and Shock Waves. 22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol.
Page ii - Perturbation Methods in Non-linear Systems. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space. 10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations. 11.
Page 397 - HW BROER, GB HUITEMA AND MB SEVRYUK, Quasi-Periodic Motions in Families of Dynamical Systems : Order amidst Chaos, Lecture Notes in Math.
Page 410 - Asy mptotics for a class of semi-linear hyperbolic equations with an application to a problem with a quadratic nonlinearity.
Page 410 - The crystal structure of cobyric acid, factor V la (with appendixes by N. Waters & Joyce Waters and Eleanor Dodson). Proc. A 323, 455-487 (1971). Vered, M. See Singh (SJ), Ben-Menahem & Vered. Verhulst, F. Discrete symmetric dynamical systems at the main resonances with applications to axisymmetric galaxies.
Page 410 - THE TRANSITION FROM ELLIPTIC TO HYPERBOLIC ORBITS IN THE TWO-BODY PROBLEM BY SLOW LOSS OF MASS.
Page 410 - WT van Horssen and AHP van der Burgh. On initial boundary value problems for weakly nonlinear telegraph equations, asymptotic theory and application.

References to this book

All Book Search results »

Bibliographic information