Elements of applied bifurcation theory:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1998
|
| Edition: | 2. ed. |
| Series: | Applied mathematical sciences
112 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 553 - 576 |
| Physical Description: | XIX, 591 S. graph. Darst. |
| ISBN: | 0387983821 |
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| 100 | 1 | |a Kuznecov, Jurij A. |d 1945- |e Verfasser |0 (DE-588)113390831 |4 aut | |
| 245 | 1 | 0 | |a Elements of applied bifurcation theory |c Yuri A. Kuznetsov |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1998 | |
| 300 | |a XIX, 591 S. |b graph. Darst. | ||
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| 490 | 1 | |a Applied mathematical sciences |v 112 | |
| 500 | |a Literaturverz. S. 553 - 576 | ||
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Record in the Search Index
| _version_ | 1804126956691652608 |
|---|---|
| adam_text | YURI A. KUZNETSOV ELEMENTS OF APPLIED BIFURCATION THEORY SECOND EDITION
WITH 251 ILLUSTRATIONS SPRINGER CONTENTS PREFACE TO THE SECOND EDITION
VII PREFACE TO THE FIRST EDITION IX 1 INTRODUCTION TO DYNAMICAL SYSTEMS
1 1.1 DEFINITION OF A DYNAMICAL SYSTEM 1 1.1.1 STATE SPACE 2 1.1.2 TIME
5 1.1.3 EVOLUTION OPERATOR 5 1.1.4 DEFINITION OF A DYNAMICAL SYSTEM 7
1.2 ORBITS AND PHASE PORTRAITS 8 1.3 INVARIANT SETS 11 1.3.1 DEFINITION
AND TYPES 11 1.3.2 EXAMPLE 1.9 (SMALE HORSESHOE) 12 1.3.3 STABILITY OF
INVARIANT SETS 16 1.4 DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS 18
1.5 POINCARE MAPS 23 1.5.1 TIME-SHIFT MAPS 24 1.5.2 POINCARE MAP AND
STABILITY OF CYCLES 25 1.5.3 POINCARE MAP FOR PERIODICALLY FORCED
SYSTEMS 30 1.6 EXERCISES 31 1.7 APPENDIX 1: INFINITE-DIMENSIONAL
DYNAMICAL SYSTEMS DEFINED BY REACTION-DIFFUSION EQUATIONS 33 1.8
APPENDIX 2: BIBLIOGRAPHICAL NOTES 37 2 TOPOLOGICAL EQUIVALENCE,
BIFURCATIONS, AND STRUCTURAL STABILITY OF DYNAMICAL SYSTEMS 39 2.1
EQUIVALENCE OF DYNAMICAL SYSTEMS 39 XVI CONTENTS 2.2 TOPOLOGICAL
CLASSIFICATION OF GENERIC EQUILIBRIA AND FIXED POINTS 46 2.2.1
HYPERBOLIC EQUILIBRIA IN CONTINUOUS-TIME SYSTEMS 46 2.2.2 HYPERBOLIC
FIXED POINTS IN DISCRETE-TIME SYSTEMS 49 2.2.3 HYPERBOLIC LIMIT CYCLES
54 2.3 BIFURCATIONS AND BIFURCATION DIAGRAMS 57 2.4 TOPOLOGICAL NORMAL
FORMS FOR BIFURCATIONS 63 2.5 STRUCTURAL STABILITY 68 2.6 EXERCISES 73
2.7 APPENDIX: BIBLIOGRAPHICAL NOTES 76 3 ONE-PARAMETER BIFURCATIONS OF
EQUILIBRIA IN CONTINUOUS-TIME DYNAMICAL SYSTEMS 79 3.1 SIMPLEST
BIFURCATION CONDITIONS 79 3.2 THE NORMAL FORM OF THE FOLD BIFURCATION 80
3.3 GENERIC FOLD BIFURCATION 83 3.4 THE NORMAL FORM OF THE HOPF
BIFURCATION 86 3.5 GENERIC HOPF BIFURCATION 91 3.6 EXERCISES 104 3.7
APPENDIX 1: PROOF OF LEMMA 3.2 108 3.8 APPENDIX 2: BIBLIOGRAPHICAL NOTES
ILL 4 ONE-PARAMETER BIFURCATIONS OF FIXED POINTS IN DISCRETE-TIME
DYNAMICAL SYSTEMS 113 4.1 SIMPLEST BIFURCATION CONDITIONS 113 4.2 THE
NORMAL FORM OF THE FOLD BIFURCATION 114 4.3 GENERIC FOLD BIFURCATION 116
4.4 THE NORMAL FORM OF THE FLIP BIFURCATION 119 4.5 GENERIC FLIP
BIFURCATION 121 4.6 THE NORMAL FORM OF THE NEIMARK-SACKER BIFURCATION
125 4.7 GENERIC NEIMARK-SACKER BIFURCATION 129 4.8 EXERCISES 138 4.9
APPENDIX 1: FEIGENBAUM S UNIVERSALITY 139 4.10 APPENDIX 2: PROOF OF
LEMMA 4.3 143 4.11 APPENDIX 3: BIBLIOGRAPHICAL NOTES 149 5 BIFURCATIONS
OF EQUILIBRIA AND PERIODIC ORBITS IN N-DIMENSIONAL DYNAMICAL SYSTEMS 151
5.1 CENTER MANIFOLD THEOREMS 151 5.1.1 CENTER MANIFOLDS IN
CONTINUOUS-TIME SYSTEMS 152 5.1.2 CENTER MANIFOLDS IN DISCRETE-TIME
SYSTEMS 156 5.2 CENTER MANIFOLDS IN PARAMETER-DEPENDENT SYSTEMS 157 5.3
BIFURCATIONS OF LIMIT CYCLES 162 5.4 COMPUTATION OF CENTER MANIFOLDS 165
5.4.1 QUADRATIC APPROXIMATION TO CENTER MANIFOLDS IN EIGENBASIS 165
5.4.2 PROJECTION METHOD FOR CENTER MANIFOLD COMPUTATION 171 CONTENTS
XVII 5.5 EXERCISES 186 5.6 APPENDIX 1: HOPF BIFURCATION IN
REACTION-DIFFUSION SYSTEMS ON THE INTERVAL WITH DIRICHLET BOUNDARY
CONDITIONS 189 5.7 APPENDIX 2: BIBLIOGRAPHICAL NOTES 193 6 BIFURCATIONS
OF ORBITS HOMOCLINIC AND HETEROCLINIC TO HYPERBOLIC EQUILIBRIA 195 6.1
HOMOCLINIC AND HETEROCLINIC ORBITS 195 6.2 ANDRONOV-LEONTOVICH THEOREM
200 6.3 HOMOCLINIC BIFURCATIONS IN THREE-DIMENSIONAL SYSTEMS: SHIL NIKOV
THEOREMS 213 6.4 HOMOCLINIC BIFURCATIONS IN N-DIMENSIONAL SYSTEMS 228
6.4.1 REGULAR HOMOCLINIC ORBITS: MELNIKOV INTEGRAL 229 6.4.2 HOMOCLINIC
CENTER MANIFOLDS 232 6.4.3 GENERIC HOMOCLINIC BIFURCATIONS IN R* 236 6.5
EXERCISES 238 6.6 APPENDIX 1: FOCUS-FOCUS HOMOCLINIC BIFURCATION IN
FOUR-DIMENSIONAL SYSTEMS 241 6.7 APPENDIX 2: BIBLIOGRAPHICAL NOTES 247 7
OTHER ONE-PARAMETER BIFURCATIONS IN CONTINUOUS-TIME DYNAMICAL SYSTEMS
249 7.1 CODIM 1 BIFURCATIONS OF HOMOCLINIC ORBITS TO NONHYPERBOLIC
EQUILIBRIA 250 7.1.1 SADDLE-NODE HOMOCLINIC BIFURCATION ON THE PLANE 250
7.1.2 SADDLE-NODE AND SADDLE-SADDLE HOMOCLINIC BIFURCATIONS IN R 3 253
7.2 EXOTIC BIFURCATIONS 262 7.2.1 NONTRANSVERSAL HOMOCLINIC ORBIT TO A
HYPERBOLIC CYCLE 263 7.2.2 HOMOCLINIC ORBITS TO A NONHYPERBOLIC LIMIT
CYCLE 263 7.3 BIFURCATIONS ON INVARIANT TORI 267 7.3.1 REDUCTION TO A
POINCARE MAP 267 7.3.2 ROTATION NUMBER AND ORBIT STRUCTURE 269 7.3.3
STRUCTURAL STABILITY AND BIFURCATIONS 270 7.3.4 PHASE LOCKING NEAR A
NEIMARK-SACKER BIFURCATION: ARNOLD TONGUES 272 7.4 BIFURCATIONS IN
SYMMETRIC SYSTEMS 276 7.4.1 GENERAL PROPERTIES OF SYMMETRIC SYSTEMS 276
7.4.2 Z 2 -EQUIVARIANT SYSTEMS 278 7.4.3 CODIM 1 BIFURCATIONS OF
EQUILIBRIA IN Z 2 -EQUIVARIANT SYSTEMS 280 7.4.4 CODIM 1 BIFURCATIONS OF
CYCLES IN Z 2 -EQUIVARIANT SYSTEMS 283 7.5 EXERCISES 288 7.6 APPENDIX 1:
BIBLIOGRAPHICAL NOTES 290 XVIII CONTENTS 8 TWO-PARAMETER BIFURCATIONS OF
EQUILIBRIA IN CONTINUOUS-TIME DYNAMICAL SYSTEMS 293 8.1 LIST OF CODIM 2
BIFURCATIONS OF EQUILIBRIA 294 8.1.1 BIFURCATION CURVES 294 8.1.2
CODIMENSION TWO BIFURCATION POINTS 297 8.2 CUSP BIFURCATION 301 8.2.1
NORMAL FORM DERIVATION 301 8.2.2 BIFURCATION DIAGRAM OF THE NORMAL FORM
303 8.2.3 EFFECT OF HIGHER-ORDER TERMS 305 8.3 BAUTIN (GENERALIZED HOPF)
BIFURCATION 307 8.3.1 NORMAL FORM DERIVATION 307 8.3.2 BIFURCATION
DIAGRAM* OF THE NORMAL FORM 312 8.3.3 EFFECT OF HIGHER-ORDER TERMS 313
8.4 BOGDANOV-TAKENS (DOUBLE-ZERO) BIFURCATION 314 8.4.1 NORMAL FORM
DERIVATION 314 8.4.2 BIFURCATION DIAGRAM OF THE NORMAL FORM 321 8.4.3
EFFECT OF HIGHER-ORDER TERMS 324 8.5 FOLD-HOPF (ZERO-PAIR) BIFURCATION
330 8.5.1 DERIVATION OF THE NORMAL FORM 330 8.5.2 BIFURCATION DIAGRAM OF
THE TRUNCATED NORMAL FORM 337 8.5.3 EFFECT OF HIGHER-ORDER TERMS 342 8.6
HOPF-HOPF BIFURCATION 349 8.6.1 DERIVATION OF THE NORMAL FORM 349 8.6.2
BIFURCATION DIAGRAM OF THE TRUNCATED NORMAL FORM 356 8.6.3 EFFECT OF
HIGHER-ORDER TERMS 366 8.7 EXERCISES 369 8.8 APPENDIX 1: LIMIT CYCLES
AND HOMOCLINIC ORBITS OF BOGDANOV NORMAL FORM 382 8.9 APPENDIX 2:
BIBLIOGRAPHICAL NOTES 390 9 TWO-PARAMETER BIFURCATIONS OF FIXED POINTS
IN DISCRETE-TIME DYNAMICAL SYSTEMS 393 9.1 LIST OF CODIM 2 BIFURCATIONS
OF FIXED POINTS 393 9.2 CUSP BIFURCATION 397 9.3 GENERALIZED FLIP
BIFURCATION 400 9.4 CHENCINER (GENERALIZED NEIMARK-SACKER) BIFURCATION
404 9.5 STRONG RESONANCES 408 9.5.1 APPROXIMATION BY A FLOW 408 9.5.2
1:1 RESONANCE 410 9.5.3 1:2 RESONANCE 415 9.5.4 1:3 RESONANCE 428 9.5.5
1:4 RESONANCE 435 9.6 CODIM 2 BIFURCATIONS OF LIMIT CYCLES 446 9.7
EXERCISES 457 9.8 APPENDIX 1: BIBLIOGRAPHICAL NOTES 460 CONTENTS XIX 10
NUMERICAL ANALYSIS OF BIFURCATIONS 463 10.1 NUMERICAL ANALYSIS AT FIXED
PARAMETER VALUES 464 10.1.1 EQUILIBRIUM LOCATION 464 10.1.2 MODIFIED
NEWTON S METHODS 466 10.1.3 EQUILIBRIUM ANALYSIS 469 10.1.4 LOCATION OF
LIMIT CYCLES 472 10.2 ONE-PARAMETER BIFURCATION ANALYSIS 478 10.2.1
CONTINUATION OF EQUILIBRIA AND CYCLES 479 10.2.2 DETECTION AND LOCATION
OF CODIM 1 BIFURCATIONS 484 10.2.3 ANALYSIS OF CODIM 1 BIFURCATIONS 488
10.2.4 BRANCHING POINTS 495 10.3 TWO-PARAMETER BIFURCATION ANALYSIS 501
10.3.1 CONTINUATION OF CODIM 1 BIFURCATIONS OF EQUILIBRIA AND FIXED
POINTS 501 10.3.2 CONTINUATION OF CODIM 1 LIMIT CYCLE BIFURCATIONS 507
10.3.3 CONTINUATION OF CODIM 1 HOMOCLINIC ORBITS 510 10.3.4 DETECTION
AND LOCATION OF CODIM 2 BIFURCATIONS 514 10.4 CONTINUATION STRATEGY 515
10.5 EXERCISES 517 10.6 APPENDIX 1: CONVERGENCE THEOREMS FOR NEWTON
METHODS 525 10.7 APPENDIX 2: DETECTION OF CODIM 2 HOMOCLINIC
BIFURCATIONS 526 10.7.1 SINGULARITIES DETECTABLE VIA EIGENVALUES 527
10.7.2 ORBIT AND INCLINATION FLIPS 529 10.7.3 SINGULARITIES ALONG
SADDLE-NODE HOMOCLINIC CURVES 534 10.8 APPENDIX 3: BIBLIOGRAPHICAL NOTES
535 A BASIC NOTIONS FROM ALGEBRA, ANALYSIS, AND GEOMETRY 541 A.I ALGEBRA
541 A.I.I MATRICES 541 A.1.2 VECTOR SPACES AND LINEAR TRANSFORMATIONS
543 A.1.3 EIGENVECTORS AND EIGENVALUES 544 A. 1.4 INVARIANT SUBSPACES,
GENERALIZED EIVENVECTORS, AND JORDAN NORMAL FORM 545 A.1.5 FREDHOLM
ALTERNATIVE THEOREM 546 A.I.6 GROUPS 546 A.2 ANALYSIS 547 A.2.1 IMPLICIT
AND INVERSE FUNCTION THEOREMS 547 A.2.2 TAYLOR EXPANSION 548 A.2.3
METRIC, NORMED, AND OTHER SPACES 549 A.3 GEOMETRY 550 A.3.1 SETS 550
A.3.2 MAPS 551 A.3.3 MANIFOLDS 551 REFERENCES 553 INDEX 577
|
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| author_GND | (DE-588)113390831 |
| author_facet | Kuznecov, Jurij A. 1945- |
| author_role | aut |
| author_sort | Kuznecov, Jurij A. 1945- |
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| building | Verbundindex |
| bvnumber | BV012330286 |
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| ctrlnum | (OCoLC)264016187 (DE-599)BVBBV012330286 |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV012330286 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:25:42Z |
| institution | BVB |
| isbn | 0387983821 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008360478 |
| oclc_num | 264016187 |
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| owner | DE-83 DE-11 DE-188 |
| owner_facet | DE-83 DE-11 DE-188 |
| physical | XIX, 591 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Kuznecov, Jurij A. 1945- Verfasser (DE-588)113390831 aut Elements of applied bifurcation theory Yuri A. Kuznetsov 2. ed. New York [u.a.] Springer 1998 XIX, 591 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 112 Literaturverz. S. 553 - 576 Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 s DE-604 Applied mathematical sciences 112 (DE-604)BV000005274 112 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008360478&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kuznecov, Jurij A. 1945- Elements of applied bifurcation theory Applied mathematical sciences Verzweigung Mathematik (DE-588)4078889-1 gnd |
| subject_GND | (DE-588)4078889-1 |
| title | Elements of applied bifurcation theory |
| title_auth | Elements of applied bifurcation theory |
| title_exact_search | Elements of applied bifurcation theory |
| title_full | Elements of applied bifurcation theory Yuri A. Kuznetsov |
| title_fullStr | Elements of applied bifurcation theory Yuri A. Kuznetsov |
| title_full_unstemmed | Elements of applied bifurcation theory Yuri A. Kuznetsov |
| title_short | Elements of applied bifurcation theory |
| title_sort | elements of applied bifurcation theory |
| topic | Verzweigung Mathematik (DE-588)4078889-1 gnd |
| topic_facet | Verzweigung Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008360478&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT kuznecovjurija elementsofappliedbifurcationtheory |