## Averaging Methods in Nonlinear Dynamical SystemsPerturbation 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 |

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### Contents

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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 |

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### Other editions - View all

Averaging Methods in Nonlinear Dynamical Systems Jan A. Sanders,Ferdinand Verhulst Limited preview - 2013 |

Averaging Methods in Nonlinear Dynamical Systems Jan A. Sanders,Ferdinand Verhulst,James Murdock No preview available - 2007 |