Bifurcation theory and methods of dynamical systems:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific
1997
|
| Schriftenreihe: | World scientific advanced series in dynamical systems
15 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 461 S. graph. Darst. |
| ISBN: | 9810220944 |
Internformat
MARC
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Datensatz im Suchindex
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| adam_text | ADVANCED SERIES IN DYNAMICAL SYSTEMS VOL. 15 BIFURCATION THEORY AND
METHODS OF DYNAMICAL SYSTEMS LUO DINGJUN DEPARTMENT OF MATHEMATICS,
NANJING NORMAL UNIVERSITY WANG XIAN DEPARTMENT OF MATHEMATICS, NANJING
UNIVERSITY ZHU DEMING DEPARTMENT OF MATHEMATICS, EAST CHINA NORMAL
UNIVERSITY HAN MAOAN DEPARTMENT OF MATHEMATICS, SHANGHAI JIAOTONG
UNIVERSITY W * WORLD SCIENTIFIC IM SINQAPORE * NEW JERSEY * L SINGAPORE
* NEW JERSEY * LONDON * HONG KONG CONTENTS PREFACE V 1 BASIC CONCEPTS
AND FACTS 1 1.1. HYPERBOLICITY AND TRANSVERSALITY. A-LEMMA 2 1.1.1.
LINEARIZATION. HYPERBOLICITY OF CRITICAL POINT. LOCAL INVARIANT
MANIFOLDS 2 1.1.2. HYPERBOLICITY OF PERIODIC SOLUTIONS. GLOBAL STABLE
AND UNSTABLE MANIFOLDS 6 1.1.3. TRANSVERSAL INTERSECTION. A-LEMMA 10
1.2. STRUCTURAL STABILITY AND BIFURCATION 14 1.2.1. STRUCTURAL STABILITY
14 1.2.2. FI-STABILITY 17 1.2.3. BIFURCATION 20 1.3. CODIMENSION AND
UNFOLDINGS 27 1.3.1. DEFINITION AND CODIMENSION-1 EXAMPLES 27 1.3.2.
BOGDANOV-TAKENS BIFURCATION 30 1.4. CENTER MANIFOLD THEORY 33 1.4.1.
MAIN THEOREMS ON CENTER MANIFOLDS 33 1.4.2. PROPERTIES OF CENTER
MANIFOLDS 37 1.4.3. CENTER MANIFOLDS DEPENDING ON PARAMETERS ... 39 2
BIFURCATION OF 2-DIMENSIONAL SYSTEMS 45 2.1. GENERALIZED HOPF
BIFURCATION 45 2.1.1. FOCAL VALUES AND MAIN THEOREM 46 2.1.2. EXAMPLES
49 2.1.3. APPLICATIONS TO HILBERT S 16TH PROBLEM 53 IX X CONTENTS 2.2.
BIFURCATION OF MULTIPLE LIMIT CYCLES 55 2.2.1. MAIN THEOREMS 55 2.2.2.
ISLAND PROBLEM 58 2.3. HOMOCLINIC AND HETEROCLINIC BIFURCATIONS 62
2.3.1. NONDEGENERATE CASE 62 2.3.2. DEGENERATE CASE 65 2.3.3. HIGHLY
DEGENERATE CASE 75 2.4. POINCARE BIFURCATION 79 2.4.1. MAIN THEOREMS 79
2.4.2. WEAKENED HILBERT S 16TH PROBLEM 82 2.4.3. TORAL TRIGONOMETRICAL
POLYNOMIAL SYSTEMS .... 83 3 BIFURCATION IN POLYNOMIAL LIENARD SYSTEMS
91 3.1. BOUNDEDNESS OF SOLUTIONS AND EXISTENCE OF LIMIT CYCLES 93 3.1.1.
CONCEPTS OF CRITICAL POINT SYSTEM 93 3.1.2. BOUNDEDNESS OF SOLUTIONS 94
3.1.3. EXISTENCE OF LIMIT CYCLES 105 3.1.4. NON-EXISTENCE CRITERIA OF
CLOSED ORBITS 106 3.2. CRITERIA FOR DECIDING THE NUMBER OF LIMIT CYCLES
. . . 110 3.2.1. PRELIMINARY LEMMAS 111 3.2.2. GENERALIZATION OF SOME
CLASSICAL THEOREMS .... 117 3.2.3. SEVERAL NEW RESULTS 122 3.3. GLOBAL
BIFURCATION OF CUBIC LIENARD SYSTEMS 132 3.3.1. GENERAL REMARKS 133
3.3.2. INTEGRABLE CASES 135 3.3.3. ONE-CRITICAL POINT CASE 138 3.3.4.
TWO-CRITICAL POINT CASE 140 3.3.5. THREE-CRITICAL POINT CASE 154 3.4.
GLOBAL BIFURCATION IN SOME APPLIED MODELS 169 3.4.1. FITZHUGH S NERVE
CONDUCTION EQUATION 170 3.4.2. A SELF-EXCITED SYSTEM 175 3.4.3.
BOGDANOV-TAKENS SYSTEM (CONTINUED) . 179 CONTENTS XI 3.4.4. UNIQUENESS
OF LIMIT CYCLES FOR QUADRATIC SYSTEM (B) AO2= O 181 4 PERIODIC PERTURBED
SYSTEMS AND INTEGRAL MANIFOLDS 185 4.1. BIFURCATION OF PERIODIC
SOLUTIONS 185 4.1.1. POINCARE MAPS AND UNIQUENESS OF PERIODIC SOLUTIONS
185 4.1.2. THE LIAPUNOV-SCHMIDT METHOD 189 4.1.3. AN ELEMENTARY METHOD
192 4.2. METHOD OF AVERAGING AND INTEGRAL MANIFOLDS 198 4.2.1. METHOD OF
AVERAGING 198 4.2.2. INTEGRAL MANIFOLDS 203 4.3. PERIODIC PERTURBED
SYSTEMS ON A PLANE 206 4.3.1. SADDLE-NODE BIFURCATION 206 4.3.2.
HYPERBOLIC AND SEMISTABLE LIMIT CYCLES 206 4.3.3. GENERIC HOPF
BIFURCATION AND LOCAL SUBHARMONIC SOLUTIONS 212 4.4. HOPF BIFURCATION OF
INVARIANT TORUS 220 4.5. POINCARE BIFURCATION OF INVARIANT TORUS 232
4.5.1. CASE A 235 4.5.2. CASE B 237 4.5.3. CASE C (GENERAL CASE) 244 5
BIFURCATIONS OF HIGHER DIMENSIONAL SYSTEMS 253 5.1. METHODS OF
BIFURCATION FUNCTIONS OF PERIODIC ORBITS 253 5.1.1. LIAPUNOV-SCHMIDT
REDUCTION AND HOPF BIFURCATION 254 5.1.2. AN ELEMENTARY BIFURCATION
METHOD 259 5.1.3. BIFURCATION AT NON-SEMISIMPLE EIGENVALUES .... 263
5.2. ZERO AND PURE IMAGINARY EIGENVALUES 271 5.2.1. BIFURCATION ANALYSIS
FOR LIMIT CYCLES 274 5.2.2. QUALITATIVE RESULTS FOR FUELL SYSTEM 282 5.3.
TWO PAIRS OF PURE IMAGINARY EIGENVALUES 283 XII CONTENTS 5.3.1. PERIODIC
ORBITS OF REDUCED PLANAR SYSTEM 284 5.3.2. EXISTENCE OF AN INVARIANT
TWO-TORUS 290 5.3.3. BIFURCATIONS OF MULTIPLE PERIODIC ORBITS 291 5.4.
GLOBAL BIFURCATIONS OF LARGE PERIODIC ORBITS 298 5.4.1. BIFURCATIONS OF
PERIODIC ORBITS 298 5.4.2. AUTONOMOUS PERTURBATIONS OF A LINEAR SYSTEM .
. 301 5.5. UNIQUENESS AND STABILITY OF BIFURCATED PERIODIC ORBITS 304
5.6. CHAOTIC DYNAMICS BIFURCATED BY SYMMETRIE HOMOCLINIC LOOPS 319
5.6.1. SYMBOLIC DYNAMICS AND SMALE HORSESHOE 319 5.6.2. CONLEY-MOSER
CONDITIONS 322 5.6.3. CHAOTIC DYNAMICS NEAR DOUBLE SYMMETRIE HOMOCLINIC
LOOPS 324 5.7. SADDLE-FOCUS HOMOCLINIC BIFURCATION. CHAOS 329 5.7.1.
NORMAL FORM AND POINCARE MAP 330 5.7.2. VERIFICATION OF CONLEY-MOSER
CONDITIONS 334 5.7.3. COMPLICATED BEHAVIOR WITH HOPF BIFURCATION . . 340
5.8. PERIOD-DOUBLING BIFURCATION IN WEAK SIL NIKOV PHENOMENON 345 5.8.1.
EXISTENCE AND STABILITY OF BIFURCATED PERIODIC ORBITS 347 5.8.2.
BIFURCATION DIAGRAM AND ASYMPTOTIC RATIO OF BIFURCATION VALUES 351
5.8.3. SUBSIDIARY HOMOCLINIC ORBITS 355 6 MELNIKOV VECTOR, HOMOCLINIC
AND HETEROCLINIC ORBITS 357 6.1. EXPONENTIAL TRICHOTOMIES 359 6.2.
MELNIKOV VECTOR IN HIGHER DIMENSIONS 372 6.2.1. A METHOD USING
ORTHOGONALITY CONDITION AND TRANSVERSALITY THEORY 373 6.2.2. A MORE
GEOMETRICAL METHOD 381 6.3. ORBITS HETEROCLINIC TO INVARIANT MANIFOLDS
390 CONTENTS XIII 6.3.1. FENICHEL S INVARIANT MANIFOLD THEORY 392 6.3.2.
EXISTENCE AND TRANSVERSALITY OF SINGULAR ORBITS . 398 6.4. HETEROCLINIC
ORBITS IN SINGULAR PERTURBATION PROBLEMS 409 6.4.1. GEOMETRIE SINGULAR
PERTURBATION THEORY 410 6.4.2. TRANSVERSAL HETEROCLINIC ORBITS 415
6.4.3. AN EXAMPLE 423 6.5. HETEROCLINIC TO NONHYPERBOLIC EQUILIBRIA 426
6.5.1. ADMISSIBLE VARIABLES AND PRINCIPAL NORMAL COORDINATES 428 6.5.2.
BIFURCATION EQUATIONS 433 6.5.3. AN EXAMPLE 440 REFERENCES 443 INDEX 457
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:32:46Z |
| institution | BVB |
| isbn | 9810220944 |
| language | English |
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| physical | XIII, 461 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
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| publisher | World Scientific |
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| series | World scientific advanced series in dynamical systems |
| series2 | World scientific advanced series in dynamical systems |
| spelling | Bifurcation theory and methods of dynamical systems Luo Dingjun ... Singapore [u.a.] World Scientific 1997 XIII, 461 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier World scientific advanced series in dynamical systems 15 Bifurcation theory Differentiable dynamical systems Dynamisches System (DE-588)4013396-5 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 s Dynamisches System (DE-588)4013396-5 s DE-604 Luo, Dingjun Sonstige oth World scientific advanced series in dynamical systems 15 (DE-604)BV000019405 15 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008659047&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bifurcation theory and methods of dynamical systems World scientific advanced series in dynamical systems Bifurcation theory Differentiable dynamical systems Dynamisches System (DE-588)4013396-5 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4078889-1 |
| title | Bifurcation theory and methods of dynamical systems |
| title_auth | Bifurcation theory and methods of dynamical systems |
| title_exact_search | Bifurcation theory and methods of dynamical systems |
| title_full | Bifurcation theory and methods of dynamical systems Luo Dingjun ... |
| title_fullStr | Bifurcation theory and methods of dynamical systems Luo Dingjun ... |
| title_full_unstemmed | Bifurcation theory and methods of dynamical systems Luo Dingjun ... |
| title_short | Bifurcation theory and methods of dynamical systems |
| title_sort | bifurcation theory and methods of dynamical systems |
| topic | Bifurcation theory Differentiable dynamical systems Dynamisches System (DE-588)4013396-5 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| topic_facet | Bifurcation theory Differentiable dynamical systems Dynamisches System Verzweigung Mathematik |
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