Oscillatory integrals and phenomena beyond all algebraic orders: with applications to homoclinic orbits in reversible systems
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2000
|
| Schriftenreihe: | Lecture notes in mathematics
1741 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 412 S. graph. Darst. |
| ISBN: | 3540677852 |
Internformat
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| 100 | 1 | |a Lombardi, Eric |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Oscillatory integrals and phenomena beyond all algebraic orders |b with applications to homoclinic orbits in reversible systems |c Eric Lombardi |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2000 | |
| 300 | |a XV, 412 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1741 | |
| 650 | 4 | |a Dynamisches System - Oszillatorisches Integral - Reversibles System - Homokliner Orbit - Soliton | |
| 650 | 4 | |a Bifurcation theory | |
| 650 | 4 | |a Differentiable dynamical systems | |
| 650 | 4 | |a Differential equations, Nonlinear |x Numerical solutions | |
| 650 | 0 | 7 | |a Oszillatorisches Integral |0 (DE-588)4610930-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Randwertproblem |0 (DE-588)4048395-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Verzweigung |g Mathematik |0 (DE-588)4078889-1 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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| adam_text | Table of Contents
1. Introduction 1
1.1 A little toy model : from phenomena beyond any algebraic
order to oscillatory integrals 1
1.2 Examples of oscillatory integrals hidden in dynamical systems 7
1.3 From mono frequency oscillatory integrals to singularities of
solutions of complex differential equations 14
1.4 From bi frequency oscillatory integrals to singularities of so¬
lutions of complex partial differential equations 16
1.5 On the contents 17
Part I. Toolbox for oscillatory integrals
2. Exponential tools for evaluating oscillatory integrals . . 23
2.1 Mono frequency oscillatory integrals: complexification of time 23
2.1.1 Rough exponential upper bounds 23
2.1.2 Sharp exponential upper bounds 29
2.1.3 Exponential equivalent : general theory 34
2.1.4 Exponential equivalent: strategy for nonlinear differ¬
ential equations 43
2.1.5 Exponential equivalent: an example 50
2.2 Bi frequency oscillatory integrals: partial complexification of
time 63
2.2.1 Exponential upper bounds and equivalent 64
2.A Appendix. Method of continuation along horizontal lines .... 74
XII Table of Contents
Part II. Toolbox for reversible systems studied near resonances
3. Resonances of reversible vector fields 79
3.1 Definitions and basic properties 79
3.1.1 Reversible vector fields 79
3.1.2 Linear classification and nomenclature of reversible
fixed points 82
3.1.3 Families of reversible vector fields and resonances 86
3.2 Global normal forms associated with resonances 87
3.2.1 The fully critical case 88
3.2.2 The general case 93
3. A Appendix. Proof of theorem of classification of reversible ma¬
trices 94
4. Analytic description of periodic orbits bifurcating from a
pair of simple purely imaginary eigenvalues 101
4.1 Real periodic orbits: explicit form 101
4.2 Complexification of the periodic orbits 105
4.3 Analytic conjugacy to circles 105
4.A Appendix. Proof of Theorem 4.1.2 106
4.A.I Rewriting of the system as an implicit equation 106
4.A.2 Study of £„ 110
4.A.3 Expansion in powers of k 114
4.B Appendix. Proof of Proposition 4.2.1 118
4.C Proof of Theorem 4.3.1 119
5. Constructive Floquet Theory for periodic matrices near a
constant one 123
5.1 Constructive Floquet Theory in the non resonant case 124
5.2 Constructive Floquet Theory in the resonant case 126
5. A Appendix. Proof of Theorem 5.1.1 127
5.A.I Writing of the equation as an implicit equation 127
5.A.2 Study of £(u) 128
5.A.3 The implicit equation 132
5.B Proof of Lemma 5.1.2 132
6. Inversion of afflne equations around reversible homoclinic
connections 135
6.1 Explicit computation of a basis of solutions of the linear ho¬
mogeneous equation 136
6.2 Complex singularities of solutions of the homogeneous equationl38
6.2.1 Theory of Fuchs for linear systems 139
6.2.2 Theory of Fuchs for equation of the nth order 142
6.3 Linearization around homoclinic connections 149
Table of Contents XIII
6.4 Affine equations with real and complexified time 158
6.5 Affine equations with partially complexified time 166
Part III. Applications to homoclinic orbits near resonances in
reversible systems
7. The 02+iu resonance 187
7.1 Introduction 187
7.1.1 Full system, normal form and scaling 189
7.1.2 Truncated system 193
7.1.3 Statement of the results of persistence for periodic so¬
lutions 195
7.1.4 Statement of the results of persistence for homoclinic
connections 196
7.2 Proof of the persistence of periodic orbits: explicit form and
complexification 207
7.3 Proof of the persistence of reversible homoclinic connections
to exponentially small periodic solutions 209
7.3.1 Choice of the parameters 211
7.3.2 Estimates of YkiV, h, N , R , p$ 213
7.3.3 Homogeneous equation 217
7.3.4 Integral equation 219
7.3.5 Choice and control of the phase shift 221
7.3.6 Fixed point 227
7.3.7 Proof of Theorem 7.1.18 229
7.4 Proof of the generic non persistence of homoclinic connections
to 0 231
7.4.1 Introduction 231
7.4.2 Local stable manifold of 0 245
7.4.3 Holomorphic continuation of the stable manifold of 0
far away from the singularities 249
7.4.4 Stable manifold of 0 for the inner system 254
7.4.5 Holomorphic continuation of the stable manifold of 0
near i?r 262
7.4.6 Symmetrization : holomorphic continuation of the sta¬
ble manifold of 0 near —i7r 267
7.4.7 Exponential asymptotics of the solvability condition .. 270
7.A Appendix. Proofs of Lemmas 7.3.9, 7.3.10, 7.3.11 279
7.B Appendix. Proof of Proposition 7.3.19 281
7.C Appendix. Proof of Proposition 7.3.24 284
7.D Appendix. Proof of Proposition 7.3.26 286
7.E Appendix. Proof of Lemma 7.3.29 287
7.F Appendix. Proof of Lemma 7.4.27 288
XIV Table of Contents
7.G Appendix. Proof of lemma 7.4.41 303
7.H Appendix. Proof of lemma 7.4.42 308
7.1 Appendix. Proof of lemma 7.4.45 312
7.J Appendix. Proof of Lemmas 7.4.15, 7.4.16 323
8. The 02+iu resonance in infinite dimensions. Application to
water waves 327
8.1 Introduction 327
8.1.1 The 02+iw resonance in infinite dimensions 327
8.1.2 Direct normalization and scaling 330
8.1.3 Truncated system 333
8.1.4 Statement of the results of persistence 333
8.2 Optimal resolution of affine equations 335
8.2.1 Green s function and classical solutions 336
8.2.2 Optimal regularity 338
8.3 Persistence of periodic orbits, explicit form and complexification342
8.3.1 Real periodic orbits : explicit form 342
8.3.2 Complexification of the periodic orbits 346
8.4 Persistence of homoclinic connections to exponentially small
periodic orbits 346
8.4.1 Choice of the space for v 347
8.4.2 Integral equation 348
8.4.3 Choice of the phase shift 349
8.4.4 Fixed point 349
8.5 Application to water waves 350
8. A Appendix. Proof of Theorem 8.1.10 353
8.A.I Substitution 353
8.A.2 Study of equation (8.28) 354
8.A.3 Study of equation (8.29) 354
8.B Appendix. Study of Cv 355
9. The (iu o)2 u* x resonance 359
9.1 Introduction 359
9.1.1 Full system, normal form and scaling 362
9.1.2 Truncated system 366
9.1.3 Statement of the results of persistence for periodic orbits369
9.1.4 Statement of the results of persistence for homoclinic
connections 370
9.2 Proof of the persistence of periodic orbits: explicit form 378
9.3 Proof of the persistence of reversible homoclinic connections
to exponentially small periodic solutions 380
9.3.1 Analytic conjugacy to circles 382
9.3.2 Equation centered on periodic orbits 384
9.3.3 Inversion of affine equation with partially complexified
time 386
Table of Contents XV
9.3.4 Partial complexification of time and choice of the pa¬
rameters 390
9.3.5 Choice and control of the phase shift 393
9.3.6 fixed point 394
9.4 Generic non persistence of reversible homoclinic connections
to 0 396
9.A Appendix. Floquet linear change of coordinates. Proof of
Proposition 9.3.2 398
References 405
Index 411
List of symbols 413
|
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| isbn | 3540677852 |
| language | English |
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| spelling | Lombardi, Eric Verfasser aut Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems Eric Lombardi Berlin [u.a.] Springer 2000 XV, 412 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1741 Dynamisches System - Oszillatorisches Integral - Reversibles System - Homokliner Orbit - Soliton Bifurcation theory Differentiable dynamical systems Differential equations, Nonlinear Numerical solutions Oszillatorisches Integral (DE-588)4610930-4 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Oszillatorisches Integral (DE-588)4610930-4 s Randwertproblem (DE-588)4048395-2 s Verzweigung Mathematik (DE-588)4078889-1 s DE-604 Lecture notes in mathematics 1741 (DE-604)BV000676446 1741 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009060911&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lombardi, Eric Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems Lecture notes in mathematics Dynamisches System - Oszillatorisches Integral - Reversibles System - Homokliner Orbit - Soliton Bifurcation theory Differentiable dynamical systems Differential equations, Nonlinear Numerical solutions Oszillatorisches Integral (DE-588)4610930-4 gnd Randwertproblem (DE-588)4048395-2 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| subject_GND | (DE-588)4610930-4 (DE-588)4048395-2 (DE-588)4078889-1 |
| title | Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems |
| title_auth | Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems |
| title_exact_search | Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems |
| title_full | Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems Eric Lombardi |
| title_fullStr | Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems Eric Lombardi |
| title_full_unstemmed | Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems Eric Lombardi |
| title_short | Oscillatory integrals and phenomena beyond all algebraic orders |
| title_sort | oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems |
| title_sub | with applications to homoclinic orbits in reversible systems |
| topic | Dynamisches System - Oszillatorisches Integral - Reversibles System - Homokliner Orbit - Soliton Bifurcation theory Differentiable dynamical systems Differential equations, Nonlinear Numerical solutions Oszillatorisches Integral (DE-588)4610930-4 gnd Randwertproblem (DE-588)4048395-2 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| topic_facet | Dynamisches System - Oszillatorisches Integral - Reversibles System - Homokliner Orbit - Soliton Bifurcation theory Differentiable dynamical systems Differential equations, Nonlinear Numerical solutions Oszillatorisches Integral Randwertproblem Verzweigung Mathematik |
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