Integrability and nonintegrability of dynamical systems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
River Edge, NJ.
World Scientific
2001
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| Schriftenreihe: | Advanced series in nonlinear dynamics
19 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 415 S. graph. Darst. |
| ISBN: | 981023533X |
Internformat
MARC
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| 245 | 1 | 0 | |a Integrability and nonintegrability of dynamical systems |c Alain Goriely |
| 264 | 1 | |a River Edge, NJ. |b World Scientific |c 2001 | |
| 300 | |a XVIII, 415 S. |b graph. Darst. | ||
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| 490 | 1 | |a Advanced series in nonlinear dynamics |v 19 | |
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| 650 | 4 | |a nonintegrability theory | |
| 650 | 4 | |a nonlinear differential equations | |
| 650 | 4 | |a nonlinear dynamics | |
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Datensatz im Suchindex
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| adam_text | A D V A N C E D S E R I E S IN N O N L I N E A R D Y N A M I C S V O L U
M E 1 9 INTEGRABILITY AND NONINTEGRABILITY OF DYNAMICAL SYSTEMS ALAIN
GORIELY UNIVERSITY OF ARIZONA, USA WORLD SCIENTIFIC NEW
JERSEY LONDON SINGAPORE* HONG KONG CONTENTS PREFACE VII CHAPTER 1
INTRODUCTION 1 1.1 A PLANAR SYSTEM 1 1.1.1 A DYNAMICAL SYSTEM APPROACH 2
1.1.2 AN ALGEBRAIC APPROACH 4 1.1.3 AN ANALYTIC APPROACH 10 1.1.4
RELEVANT QUESTIONS 13 1.2 THE LORENZ SYSTEM 14 1.2.1 A DYNAMICAL SYSTEM
APPROACH 15 1.2.2 AN ALGEBRAIC APPROACH 16 1.2.3 AN ANALYTIC APPROACH 19
1.2.4 RELEVANT QUESTIONS 22 1.3 EXERCISES 23 CHAPTER 2 INTEGRABILITY: AN
ALGEBRAIC APPROACH 25 2.1 FIRST INTEGRALS 26 2.1.1 A CANONICAL EXAMPLE:
THE RIGID BODY MOTION 32 2.2 CLASSES OF FUNCTIONS 35 2.2.1 ELEMENTARY
FIRST INTEGRALS 38 2.2.2 DIFFERENTIAL FIELDS 40 2.3 HOMOGENEOUS VECTOR
FIELDS 42 2.3.1 SCALE-INVARIANT SYSTEMS 44 2.3.2 HOMOGENEOUS AND
WEIGHT-HOMOGENEOUS DECOMPOSITIONS 46 2.3.3 WEIGHT-HOMOGENEOUS
DECOMPOSITIONS 47 XIII XIV CONTENTS 2.4 BUILDING FIRST INTEGRALS 49
2.4.1 A SIMPLE ALGORITHM FOR POLYNOMIAL FIRST INTEGRALS . . . . 49 2.5
SECOND INTEGRALS 51 2.5.1 DARBOUX POLYNOMIALS 54 2.5.2 DARBOUX
POLYNOMIALS FOR PLANAR VECTOR FIELDS 55 2.5.3 THE PRELLE-SINGER
ALGORITHM 57 2.6 THIRD INTEGRALS 60 2.7 HIGHER INTEGRALS 63 2.8
CLASS-REDUCTION 64 2.9 FIRST INTEGRALS FOR VECTOR FIELDS IN M 3 : THE
COMPATIBILITY ANALYSIS 67 2.10 INTEGRABILITY 71 2.10.1 LOCAL
INTEGRABILITY 71 2.10.2 LIOUVILLE INTEGRABILITY 72 2.10.3 ALGEBRAIC
INTEGRABILITY 73 2.11 JACOBI S LAST MULTIPLIER METHOD 74 2.12 LAX PAIRS
79 2.12.1 GENERAL PROPERTIES 80 2.12.2 CONSTRUCTION OF LAX PAIRS 91
2.12.3 COMPLETION OF LAX PAIRS 94 2.12.4 RECYCLING INTEGRABLE SYSTEMS 97
2.12.5 MORE ON LAX PAIRS 99 2.13 EXERCISES 100 CHAPTER 3 INTEGRABILITY:
AN ANALYTIC APPROACH 105 3.1 SINGULARITIES OF FUNCTIONS 106 3.2
SOLUTIONS OF DIFFERENTIAL EQUATIONS 108 3.3 SINGULARITIES OF LINEAR
DIFFERENTIAL EQUATIONS 110 3.3.1 FUNDAMENTAL SOLUTIONS ILL 3.3.2 REGULAR
SINGULAR POINTS 112 3.4 SINGULARITIES OF NONLINEAR DIFFERENTIAL
EQUATIONS 113 3.4.1 FIXED AND MOVABLE SINGULARITIES 113 3.5 THE PAINLEVE
PROPERTY 114 3.5.1 HISTORICAL DIGRESSION I 115 3.5.2 PAINLEVE S A-METHOD
119 3.5.3 THE ISOMONODROMY DEFORMATION PROBLEM 121 3.5.4 APPLICATIONS
121 3.6 PAINLEVE EQUATIONS AND INTEGRABLE PDES 122 3.6.1 THE THEORY OF
SOLITONS AND THE INVERSE SCATTERING TRANSFORML23 CONTENTS XV 3.6.2 THE
ABLOWITZ-RAMANI-SEGUR CONJECTURE 124 3.7 THE PDE PAINLEVE TEST 125 3.7.1
INTEGRABILITY OF ODES 128 3.8 SINGULARITY ANALYSIS 129 3.8.1 STEP 1: THE
DOMINANT BEHAVIOR 131 3.8.2 STEP 2: KOVALEVSKAYA EXPONENTS 134 3.8.3
STEP 3: THE LOCAL SOLUTION 138 3.8.3.1 FORMAL SERIES 138 3.8.3.2 THE
PUISEUX SERIES 139 3.8.3.3 LOGARITHMIC EXPANSIONS 140 3.8.4 FORMAL
EXISTENCE OF LOCAL SOLUTIONS 143 3.8.5 COMPANION SYSTEMS 146 3.8.5.1
LOCAL ANALYSIS AROUND FIXED POINTS 147 3.8.5.2 COMPANION TRANSFORMATION
148 3.8.5.3 SINGULARITY ANALYSIS AND UNSTABLE MANIFOLDS . 150 3.8.6
CONVERGENCE OF LOCAL SOLUTIONS 151 3.8.7 A SHORT LIST OF SINGULARITY
ANALYSES 153 3.9 THE PAINLEVE TESTS 155 3.9.1 PAINLEVE TEST #1: THE
HOYER-KOVALEVSKAYA METHOD ... 155 3.9.1.1 HISTORICAL DIGRESSION II 155
3.9.1.2 THE KOVALEVSKAYA-HOYER PROCEDURE 157 3.9.2 PAINLEVE TEST #2: THE
GAMBIER-ARS ALGORITHM 163 3.9.2.1 THE ALGORITHM 164 3.9.3 PAINLEVE TEST
#3: THE PAINLEVE-CFP ALGORITHM 167 3.9.3.1 THE ALGORITHMS 169 3.9.4
PAINLEVE PROPERTY AND NORMAL FORMS 171 3.9.4.1 NORMAL FORMS OF ANALYTIC
VECTOR FIELDS 172 3.9.4.2 PAINLEVE PROPERTY AND LINEARIZABILITY . . . .
. 175 3.9.4.3 YET ANOTHER ALGORITHM 178 3.9.4.4 A DIGRESSION: THE
PROBLEM OF THE CENTER. . . . 185 3.10 THE WEAK-PAINLEVE CONJECTURE 185
3.11 PATTERNS OF SINGULARITIES FOR NONINTEGRABLE SYSTEMS 188 3.11.1
KOVALEVSKAYA FRACTALS 188 3.11.2 SINGULARITY CLUSTERING 190 3.12 FINITE
TIME BLOW-UP : 190 3.13 EXERCISES 201 XVI CONTENTS CHAPTER 4 POLYNOMIAL
AND QUASI-POLYNOMIAL VECTOR FIELDS 207 4.1 THE QUASIMONOMIAL SYSTEMS 208
4.2 THE QUASIMONOMIAL TRANSFORMATIONS 210 4.3 NEW-TIME TRANSFORMATIONS
212 4.4 CANONICAL FORMS 214 4.5 THE NEWTON POLYHEDRON 215 4.6
TRANSFORMATION OF THE NEWTON POLYHEDRON 217 4.7 HISTORICAL DIGRESSION: A
NEW-OLD FORMALISM 220 4.8 ALGEBRAIC DEGENERACY 222 4.8.1 DEGENERACY OF
MATRIX A 223 4.8.2 DEGENERACY OF MATRIX B 225 4.9 TRANSFORMATION OF
FIRST INTEGRALS 225 4.10 AN ALGORITHM FOR POLYNOMIAL FIRST INTEGRALS 226
4.11 JACOBI S LAST MULTIPLIER FOR QUASIMONOMIAL SYSTEMS 229 4.12
APPLICATION: SEMI-SIMPLE NORMAL FORMS 230 4.13 QUASIMONOMIAL
TRANSFORMATION AND THE PAINLEVE PROPERTY . . . 232 4.13.1 THE
TRANSFORMATIONS GROUP OF THE RIEMANN SPHERE . . . 233 4.14 PAINLEVE
TESTS AND QUASIMONOMIAL TRANSFORMATIONS 234 4.14.1 THE DOMINANT BALANCES
235 4.14.2 DOMINANT BALANCE AND NEWTON S POLYHEDRON 238 4.14.3 THE
KOVALEVSKAYA EXPONENTS 240 4.14.4 QUASIMONOMIAL TRANSFORMATION OF LOCAL
SERIES 241 4.15 THE PAINLEVE TEST FOR THE LOTKA-VOLTERRA FORM 245 4.15.1
STEP 1: DOMINANT BALANCES 245 4.15.2 STEP 2: KOVALEVSKAYA EXPONENTS 246
4.15.3 STEP 3: COMPATIBILITY CONDITIONS 246 4.16 TRANSFORMATION OF
SINGULARITIES 248 4.16.1 NEW-TIME TRANSFORMATION OF LOCAL SERIES 249
4.16.2 THE WEAK-PAINLEVE CONJECTURE 252 4.16.3 NEW INTEGRABLE SYSTEMS
256 4.17 EXERCISES 259 CHAPTER 5 NONINTEGRABILITY 263 5.1 THE GENERAL
APPROACH: THE VARIATIONAL EQUATION 264 5.1.1 NONINTEGRABILITY OF LINEAR
SYSTEMS 267 5.2 FIRST INTEGRALS AND LINEAR EIGENVALUES 269 5.3 FIRST
INTEGRALS AND KOVALEVSKAYA EXPONENTS 272 5.3.1 YOSHIDA S ANALYSIS 272
CONTENTS XVII 5.3.2 RESONANCES BETWEEN KOVALEVSKAYA EXPONENTS 277 5.3.3
KOVALEVSKAYA EXPONENTS AND DARBOUX POLYNOMIALS . . . 282 5.3.4
KOVALEVSKAYA EXPONENTS FOR HAMILTONIAN SYSTEMS .... 284 5.4 COMPLETE
INTEGRABILITY AND RESONANCES 284 5.5 COMPLETE INTEGRABILITY AND
LOGARITHMIC BRANCH POINTS 286 5.6 MULTIVALUED FIRST INTEGRAL AND LOCAL
SOLUTIONS 288 5.7 PARTIAL INTEGRABILITY R 291 5.7.1 A NATURAL ARBITRARY
PARAMETER 291 5.7.2 NECESSARY CONDITIONS FOR PARTIAL INTEGRABILITY 292
5.8 EXERCISES 297 CHAPTER 6 HAMILTONIAN SYSTEMS 301 6.1 HAMILTONIAN
SYSTEMS 301 6.1.1 FIRST INTEGRALS 305 6.2 COMPLETE INTEGRABILITY 309
6.2.1 LIOUVILLE INTEGRABILITY 309 6.2.2 ARNOLD-LIOUVILLE INTEGRABILITY
310 6.3 ALGEBRAIC INTEGRABILITY 312 6.4 ZIGLIN S THEORY OF
NONINTEGRABILITY 315 6.4.1 HAMILTONIAN SYSTEMS WITH TWO DEGREES OF
FREEDOM . . . 321 6.4.1.1 HOMOGENEOUS POTENTIALS 323 6.4.2 ZIGLIN S
THEOREM IN N DIMENSIONS 325 6.4.3 MORE ON ZIGLIN S THEORY 327 6.4.4 THE
MORALES-RUIZ AND RAMIS THEOREM 328 6.5 EXERCISES 330 CHAPTER 7 NEARLY
INTEGRABLE DYNAMICAL SYSTEMS 333 7.0.1 AN INTRODUCTORY EXAMPLE 335 7.1
GENERAL SETUP 339 7.1.1 GENERAL ASSUMPTIONS 340 7.1.2 COMMENTS ON THE
ASSUMPTIONS 341 7.2 A PERTURBATIVE SINGULARITY ANALYSIS 342 7.2.1 THE
PAINLEVE TEST 342 7.2.2 THE ^-SERIES 343 7.2.3 EPSILON-EXPANSION FOR THE
^-SERIES . 344 7.3 THE MELNIKOV VECTOR IN N DIMENSIONS 347 7.3.1 THE
VARIATIONAL EQUATION 348 7.3.2 THE MELNIKOV VECTOR . 351 XVIII CONTENTS
7.3.3 THE METHOD OF RESIDUES 352 7.4 SINGULARITY ANALYSIS AND THE
MELNIKOV VECTOR 355 7.4.1 THE FUNDAMENTAL LOCAL SOLUTION 356 7.4.2
RESIDUES AND LOCAL SOLUTIONS 358 7.4.3 THE COMPATIBILITY CONDITIONS AND
THE RESIDUES 359 7.5 THE ALGORITHMIC PROCEDURE 362 7.5.1 THE COMPUTATION
OF THE MELNIKOV VECTOR: ANON-ALGORITHMIC PROCEDURE 362 7.5.2 THE
ALGORITHMIC PROCEDURE 362 7.6 SOME ILLUSTRATIVE EXAMPLES 364 7.7
EXERCISES 373 CHAPTER 8 OPEN PROBLEMS 375 GLOSSARY 381 BIBLIOGRAPHY 385
INDEX 410
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:32:42Z |
| institution | BVB |
| isbn | 981023533X |
| language | English |
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| physical | XVIII, 415 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | World Scientific |
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| series | Advanced series in nonlinear dynamics |
| series2 | Advanced series in nonlinear dynamics |
| spelling | Goriely, Alain Verfasser aut Integrability and nonintegrability of dynamical systems Alain Goriely River Edge, NJ. World Scientific 2001 XVIII, 415 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Advanced series in nonlinear dynamics 19 dynamical systems integrability theory mathematics nonintegrability theory nonlinear differential equations nonlinear dynamics nonlinearity Dynamisches System (DE-588)4013396-5 gnd rswk-swf Integrierbarkeit (DE-588)4474751-2 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Integrierbarkeit (DE-588)4474751-2 s DE-604 Advanced series in nonlinear dynamics 19 (DE-604)BV004464593 19 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008656460&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Goriely, Alain Integrability and nonintegrability of dynamical systems Advanced series in nonlinear dynamics dynamical systems integrability theory mathematics nonintegrability theory nonlinear differential equations nonlinear dynamics nonlinearity Dynamisches System (DE-588)4013396-5 gnd Integrierbarkeit (DE-588)4474751-2 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4474751-2 |
| title | Integrability and nonintegrability of dynamical systems |
| title_auth | Integrability and nonintegrability of dynamical systems |
| title_exact_search | Integrability and nonintegrability of dynamical systems |
| title_full | Integrability and nonintegrability of dynamical systems Alain Goriely |
| title_fullStr | Integrability and nonintegrability of dynamical systems Alain Goriely |
| title_full_unstemmed | Integrability and nonintegrability of dynamical systems Alain Goriely |
| title_short | Integrability and nonintegrability of dynamical systems |
| title_sort | integrability and nonintegrability of dynamical systems |
| topic | dynamical systems integrability theory mathematics nonintegrability theory nonlinear differential equations nonlinear dynamics nonlinearity Dynamisches System (DE-588)4013396-5 gnd Integrierbarkeit (DE-588)4474751-2 gnd |
| topic_facet | dynamical systems integrability theory mathematics nonintegrability theory nonlinear differential equations nonlinear dynamics nonlinearity Dynamisches System Integrierbarkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008656460&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004464593 |
| work_keys_str_mv | AT gorielyalain integrabilityandnonintegrabilityofdynamicalsystems |