Bifurcation and chaos in discontinuous and continuous systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Beijing
Higher Education Press
2011
Berlin ; Heidelberg Springer |
| Series: | Nonlinear physical science
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XII, 378 S. graph. Darst. 24 cm |
| ISBN: | 3642182682 9783642182686 9787040315332 |
Staff View
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Record in the Search Index
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| adam_text |
IMAGE 1
CONTENTS
1 INTRODUCTION 1
REFERENCES 6
2 PRELIMINARY RESULTS 9
2.1 LINEAR FUNCTIONAL ANALYSIS 9
2.2 NONLINEAR FUNCTIONAL ANALYSIS 11
2.2.1 BANACH FIXED POINT THEOREM 11
2.2.2 IMPLICIT FUNCTION THEOREM 11
2.2.3 LYAPUNOV-SCHMIDT METHOD 12
2.2.4 BROUWER DEGREE 13
2.2.5 LOCAL INVERTIBILITY 13
2.2.6 GLOBAL INVERTIBILITY 14
2.3 MULTIVALUED MAPPINGS 14
2.4 DIFFERENTIAL TOPOLOGY 15
2.4.1 DIFFERENTIABLE MANIFOLDS 15
2.4.2 VECTOR BUNDLES 16
2.4.3 TUBULAR NEIGHBOURHOODS 16
2.5 DYNAMICAL SYSTEMS 17
2.5.1 HOMOGENOUS LINEAR EQUATIONS 17
2.5.2 CHAOS IN DIFFEOMORPHISMS 18
2.5.3 PERIODIC ODES 19
2.5.4 VECTOR FIELDS 20
2.5.5 GLOBAL CENTER MANIFOLDS 22
2.5.6 TWO-DIMENSIONAL FLOWS 22
2.5.7 AVERAGING METHOD 23
2.5.8 CARATHEODORY TYPE ODES 24
2.6 SINGULARITIES OF SMOOTH MAPS 24
2.6.1 JET BUNDLES 24
2.6.2 WHITNEY C TOPOLOGY 25
2.6.3 TRANSVERSALITY 25
2.6.4 MALGRANGE PREPARATION THEOREM 26
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1008972169
DIGITALISIERT DURCH
IMAGE 2
CONTENTS
2.6.5 COMPLEX ANALYSIS 26
REFERENCES 28
CHAOS IN DISCRETE DYNAMICAL SYSTEMS 29
3.1 TRANSVERSAL BOUNDED SOLUTIONS 29
3.1.1 DIFFERENCE EQUATIONS 29
3.1.2 VARIATIONAL EQUATION 30
3.1.3 PERTURBATION THEORY 35
3.1.4 BIFURCATION FROM A MANIFOLD OF HOMOCLINIC SOLUTIONS 38 3.1.5
APPLICATIONS TO IMPULSIVE DIFFERENTIAL EQUATIONS 40 3.2 TRANSVERSAL
HOMOCLINIC ORBITS 44
3.2.1 HIGHER DIMENSIONAL DIFFERENCE EQUATIONS 44
3.2.2 BIFURCATION RESULT 45
3.2.3 APPLICATIONS TO MCMILLAN TYPE MAPPINGS 51
3.2.4 PLANAR INTEGRABLE MAPS WITH SEPARATRICES 54
3.3 SINGULAR IMPULSIVE ODES 55
3.3.1 SINGULAR ODES WITH IMPULSES 55
3.3.2 LINEAR SINGULAR ODES WITH IMPULSES 56
3.3.3 DERIVATION OF THE MELNIKOV FUNCTION 64
3.3.4 EXAMPLES OF SINGULAR IMPULSIVE ODES 68
3.4 SINGULARLY PERTURBED IMPULSIVE ODES 70
3.4.1 SINGULARLY PERTURBED ODES WITH IMPULSES 70
3.4.2 MELNIKOV FUNCTION 71
3.4.3 SECOND ORDER SINGULARLY PERTURBED ODES WITH IMPULSES . . 72 3.5
INFLATED DETERMINISTIC CHAOS 73
3.5.1 INFLATED DYNAMICAL SYSTEMS 73
3.5.2 INFLATED CHAOS 74
REFERENCES 83
CHAOS IN ORDINARY DIFFERENTIAL EQUATIONS 87
4.1 HIGHER DIMENSIONAL ODES 87
4.1.1 PARAMETERIZED HIGHER DIMENSIONAL ODES 87
4.1.2 VARIATIONAL EQUATIONS 88
4.1.3 MELNIKOV MAPPINGS 90
4.1.4 THE SECOND ORDER MELNIKOV FUNCTION 93
4.1.5 APPLICATION TO PERIODICALLY PERTURBED ODES 95
4.2 ODES WITH NONRESONANT CENTER MANIFOLDS 97
4.2.1 PARAMETERIZED COUPLED OSCILLATORS 97
4.2.2 CHAOTIC DYNAMICS ON THE HYPERBOLIC SUBSPACE 98 4.2.3 CHAOS IN THE
FULL EQUATION 100
4.2.4 APPLICATIONS TO NONLINEAR ODES 105
4.3 ODES WITH RESONANT CENTER MANIFOLDS 108
4.3.1 ODES WITH SADDLE-CENTER PARTS 108
4.3.2 EXAMPLE OF COUPLED OSCILLATORS AT RESONANCE 109 4.3.3 GENERAL
EQUATIONS 121
IMAGE 3
CONTENTS
4.3.4 AVERAGING METHOD 127
4.4 SINGULARLY PERTURBED AND FORCED ODES 131
4.4.1 FORCED SINGULAR ODES 131
4.4.2 CENTER MANIFOLD REDUCTION 132
4.4.3 ODES WITH NORMAL AND SLOW VARIABLES 135
4.4.4 HOMOCLINIC HOPF BIFURCATION 135
4.5 BIFURCATION FROM DEGENERATE HOMOCLINICS 136
4.5.1 PERIODICALLY FORCED ODES WITH DEGENERATE HOMOCLINICS. 136 4.5.2
BIFURCATION EQUATION 137
4.5.3 BIFURCATION FOR 2-PARAMETRIC SYSTEMS 138
4.5.4 BIFURCATION FOR 4-PARAMETRIC SYSTEMS 144
4.5.5 AUTONOMOUS PERTURBATIONS 147
4.6 INFLATED ODES 150
4.6.1 INFLATED CARATHEODORY TYPE ODES 150
4.6.2 INFLATED PERIODIC ODES 151
4.6.3 INFLATED AUTONOMOUS ODES 154
4.7 NONLINEAR DIATOMIC LATTICES 156
4.7.1 FORCED AND COUPLED NONLINEAR LATTICES 156
4.7.2 SPATIALLY LOCALIZED CHAOS 157
REFERENCES 163
CHAOS IN PARTIAL DIFFERENTIAL EQUATIONS 167
5.1 BEAMS ON ELASTIC BEARINGS 167
5.1.1 WEAKLY NONLINEAR BEAM EQUATION 167
5.1.2 SETTING OF THE PROBLEM 168
5.1.3 PRELIMINARY RESULTS 171
5.1.4 CHAOTIC SOLUTIONS 191
5.1.5 USEFUL NUMERICAL ESTIMATES 215
5.1.6 LIPSCHITZ CONTINUITY 217
5.2 INFINITE DIMENSIONAL NON-RESONANT SYSTEMS 220
5.2.1 BUCKLED ELASTIC BEAM 220
5.2.2 ABSTRACT PROBLEM 224
5.2.3 CHAOS ON THE HYPERBOLIC SUBSPACE 224
5.2.4 CHAOS IN THE FULL EQUATION 226
5.2.5 APPLICATIONS TO VIBRATING ELASTIC BEAMS 227
5.2.6 PLANER MOTION WITH ONE BUCKLED MODE 227
5.2.7 NONPLANER SYMMETRIC BEAMS 230
5.2.8 NONPLANER NONSYMMETRIC BEAMS 235
5.2.9 MULTIPLE BUCKLED MODES 238
5.3 PERIODICALLY FORCED COMPRESSED BEAM 242
5.3.1 RESONANT COMPRESSED EQUATION 242
5.3.2 FORMULATION OF WEAK SOLUTIONS 242
5.3.3 CHAOTIC SOLUTIONS 243
REFERENCES 247
IMAGE 4
XII CONTENTS
6 CHAOS IN DISCONTINUOUS DIFFERENTIAL EQUATIONS 249
6.1 TRANSVERSAL HOMOCLINIC BIFURCATION 249
6.1.1 DISCONTINUOUS DIFFERENTIAL EQUATIONS 249
6.1.2 SETTING OF THE PROBLEM 250
6.1.3 GEOMETRIC INTERPRETATION OF NONDEGENERACY CONDITION 255 6.1.4
ORBITS CLOSE TO THE LOWER HOMOCLINIC BRANCHES 257 6.1.5 ORBITS CLOSE TO
THE UPPER HOMOCLINIC BRANCH 263 6.1.6 BIFURCATION EQUATION 265
6.1.7 CHAOTIC BEHAVIOUR 287
6.1.8 ALMOST AND QUASIPERIODIC CASES 293
6.1.9 PERIODIC CASE 294
6.1.10 PIECEWISE SMOOTH PLANAR SYSTEMS 295
6.1.11 3D QUASIPERIODIC PIECEWISE LINEAR SYSTEMS 299 6.1.12 MULTIPLE
TRANSVERSAL CROSSINGS 310
6.2 SLIDING HOMOCLINIC BIFURCATION 312
6.2.1 HIGHER DIMENSIONAL SLIDING HOMOCLINICS 312
6.2.2 PLANAR SLIDING HOMOCLINICS 319
6.2.3 THREE-DIMENSIONAL SLIDING HOMOCLINICS 321
6.3 OUTLOOK 332
REFERENCES 332
7 CONCLUDING RELATED TOPICS 335
7.1 NOTES ON MELNIKOV FUNCTION 335
7.1.1 ROLE OF MELNIKOV FUNCTION 335
7.1.2 MELNIKOV FUNCTION AND CALCULUS OF RESIDUES 336 7.1.3 SECOND ORDER
ODES 340
7.1.4 APPLICATIONS AND EXAMPLES 347
7.2 TRANSVERSE HETEROCLINIC CYCLES 361
7.3 BLUE SKY CATASTROPHES 369
7.3.1 SYMMETRIC SYSTEMS WITH FIRST INTEGRALS 370
7.3.2 D'ALEMBERT AND PENALIZED EQUATIONS 371
REFERENCES 373
INDEX 375 |
| any_adam_object | 1 |
| author | Fečkan, Michal |
| author_GND | (DE-588)136478395 |
| author_facet | Fečkan, Michal |
| author_role | aut |
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| ctrlnum | (OCoLC)748688616 (DE-599)DNB1008972169 |
| dewey-full | 515.39 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.39 |
| dewey-search | 515.39 |
| dewey-sort | 3515.39 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
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| indexdate | 2024-08-03T00:45:38Z |
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| isbn | 3642182682 9783642182686 9787040315332 |
| language | English |
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| physical | XII, 378 S. graph. Darst. 24 cm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
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| publisher | Higher Education Press Springer |
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| series2 | Nonlinear physical science |
| spelling | Fečkan, Michal Verfasser (DE-588)136478395 aut Bifurcation and chaos in discontinuous and continuous systems Michal Fečkan Beijing Higher Education Press 2011 Berlin ; Heidelberg Springer XII, 378 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Nonlinear physical science Literaturangaben Chaostheorie (DE-588)4009754-7 gnd rswk-swf Nichtlineares dynamisches System (DE-588)4126142-2 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Nichtlineares dynamisches System (DE-588)4126142-2 s Chaostheorie (DE-588)4009754-7 s Verzweigung Mathematik (DE-588)4078889-1 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-18269-3 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3631538&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026087369&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fečkan, Michal Bifurcation and chaos in discontinuous and continuous systems Chaostheorie (DE-588)4009754-7 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| subject_GND | (DE-588)4009754-7 (DE-588)4126142-2 (DE-588)4078889-1 |
| title | Bifurcation and chaos in discontinuous and continuous systems |
| title_auth | Bifurcation and chaos in discontinuous and continuous systems |
| title_exact_search | Bifurcation and chaos in discontinuous and continuous systems |
| title_full | Bifurcation and chaos in discontinuous and continuous systems Michal Fečkan |
| title_fullStr | Bifurcation and chaos in discontinuous and continuous systems Michal Fečkan |
| title_full_unstemmed | Bifurcation and chaos in discontinuous and continuous systems Michal Fečkan |
| title_short | Bifurcation and chaos in discontinuous and continuous systems |
| title_sort | bifurcation and chaos in discontinuous and continuous systems |
| topic | Chaostheorie (DE-588)4009754-7 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| topic_facet | Chaostheorie Nichtlineares dynamisches System Verzweigung Mathematik |
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