Hyperbolic chaos: a physicist's view
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Beijing
Higher Education Press
2012
Berlin Springer |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XV, 320 S. Ill., graph. Darst. |
| ISBN: | 9783642236662 9787040319644 9783642236655 3642236650 |
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Record in the Search Index
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IMAGE 1
CONTENTS
PART I BASIC NOTIONS AND REVIEW 1 DYNAMICAL SYSTEMS AND HYPERBOLICITY 3
1.1 DYNAMICAL SYSTEMS: BASIC NOTIONS 3
1.1.1 SYSTEMS WITH CONTINUOUS AND DISCRETE TIME, AND THEIR MUTUAL
RELATION 3
1.1.2 DYNAMICS IN TERMS OF PHASE FLUID: CONSERVATIVE AND DISSIPATIVE
SYSTEMS AND ATTRACTORS 6
1.1.3 ROUGH SYSTEMS AND STRUCTURAL STABILITY 8
1.1.4 LYAPUNOV EXPONENTS AND THEIR COMPUTATION 10
1.2 MODEL EXAMPLES OF CHAOTIC ATTRACTORS 12
1.2.1 CHAOS IN TERMS OF PHASE FLUID AND BAKER'S MAP 12
1.2.2 SMALE-WILLIAMS SOLENOID 15
1.2.3 DA-ATTRACTOR 16
1.2.4 PLYKIN TYPE ATTRACTORS 17
1.3 NOTION OF HYPERBOLICITY 19
1.4 CONTENT AND CONCLUSIONS OF THE HYPERBOLIC THEORY 22
1.4.1 CONE CRITERION 24
1.4.2 INSTABILITY 25
1.4.3 TRANSVERSAL CANTOR STRUCTURE AND KAPLAN-YORKE DIMENSION 25
1.4.4 MARKOV PARTITION AND SYMBOLIC DYNAMICS 26
1.4.5 ENUMERATING OF ORBITS AND TOPOLOGICAL ENTROPY 27 1.4.6 STRUCTURAL
STABILITY 28
1.4.7 INVARIANT MEASURE OF SINAI-RUELLE-BOWEN 29
1.4.8 SHADOWING AND EFFECT OF NOISE 30
1.4.9 ERGODICITY AND MIXING 30
1.4.10 KOLMOGOROV-SINAI ENTROPY 31
REFERENCES 31
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1014072662
DIGITALISIERT DURCH
IMAGE 2
JYJ CONTENTS
2 POSSIBLE OCCURRENCE OF HYPERBOLIC ATTRACTORS 35
2.1 THE NEWHOUSE-RUELLE-TAKENS THEOREM AND ITS RELATION TO THE UNIFORMLY
HYPERBOLIC ATTRACTORS 35
2.2 LORENZ MODEL AND ITS MODIFICATIONS 37
2.3 SOME MAPS WITH UNIFORMLY HYPERBOLIC ATTRACTORS 40
2.4 FROM DA TO THE PLYKIN TYPE ATTRACTOR 43
2.5 HUNT'S EXAMPLE: SUSPENDING THE PLYKIN TYPE ATTRACTOR 46 2.6 THE
TRIPLE LINKAGE: A MECHANICAL SYSTEM WITH HYPERBOLIC DYNAMICS 49
2.7 A POSSIBLE OCCURRENCE OF A PLYKIN TYPE ATTRACTOR IN HINDMARSH-ROSE
NEURON MODEL 51
2.8 BLUE SKY CATASTROPHE AND BIRTH OF THE SMALE-WILLIAMS ATTRACTOR 52
2.9 TAFFY-PULLING MACHINE 53
REFERENCES 54
PART II LOW-DIMENSIONAL MODELS
3 KICKED MECHANICAL MODELS AND DIFFERENTIAL EQUATIONS WITH PERIODIC
SWITCH 59
3.1 SMALE-WILLIAMS SOLENOID IN MECHANICAL MODEL: MOTION OF A PARTICLE ON
A PLANE UNDER PERIODIC KICKS 60
3.2 A SET OF SWITCHING DIFFERENTIAL EQUATIONS WITH ATTRACTOR OF
SMALE-WILLIAMS TYPE 65
3.3 EXPLICIT DYNAMICAL SYSTEM WITH ATTRACTOR OF PLYKIN TYPE 68 3.3.1
PLYKIN TYPE ATTRACTOR ON A SPHERE 68
3.3.2 PLYKIN TYPE ATTRACTOR ON THE PLANE 73
3.4 PLYKIN-LIKE ATTRACTOR IN SMOOTH NON-AUTONOMOUS SYSTEM 76 REFERENCES
79
4 NON-AUTONOMOUS SYSTEMS OF COUPLED SELF-OSCILLATORS 81 4.1 VAN DER POL
OSCILLATOR 81
4.2 SMALE-WILLIAMS ATTRACTOR IN A NON-AUTONOMOUS SYSTEM OF ALTERNATELY
EXCITED VAN DER POL OSCILLATORS 84
4.3 SYSTEM OF ALTERNATELY EXCITED VAN DER POL OSCILLATORS IN TERMS OF
SLOW COMPLEX AMPLITUDES 93
4.4 NON-RESONANCE EXCITATION TRANSFER 94
4.5 PLYKIN-LIKE ATTRACTOR IN NON-AUTONOMOUS COUPLED OSCILLATORS 95 4.5.1
REPRESENTATION OF STATES ON A SPHERE AND EQUATIONS OF THE MODEL 95
4.5.2 NUMERICAL RESULTS FOR THE COUPLED OSCILLATORS 98
REFERENCES 101
5 AUTONOMOUS LOW-DIMENSIONAL SYSTEMS WITH UNIFORMLY HYPERBOLIC
ATTRACTORS IN THE POINCARE MAPS 103
5.1 AUTONOMOUS SYSTEM OF TWO COUPLED OSCILLATORS WITH SELF-REGULATING
ALTERNATING EXCITATION 103
IMAGE 3
CONTENTS XIII
5.2 SYSTEM CONSTRUCTED ON A BASE OF THE PREDATOR-PREY MODEL 107 5.3
EXAMPLE OF BLUE SKY CATASTROPHE ACCOMPANIED BY A BIRTH OF SMALE-WILLIAMS
ATTRACTOR 112
REFERENCES 117
6 PARAMETRIC GENERATORS OF HYPERBOLIC CHAOS 119
6.1 PARAMETRIC EXCITATION OF COUPLED OSCILLATORS. THREE-FREQUENCY
PARAMETRIC GENERATOR AND ITS OPERATION 120
6.2 HYPERBOLIC CHAOS IN PARAMETRIC OSCILLATOR WITH Q-SWITCH AND PUMP
MODULATION 123
6.2.1 DYNAMICAL EQUATIONS 123
6.2.2 QUALITATIVE EXPLANATION OF THE OPERATION 126
6.2.3 NUMERICAL RESULTS 127
6.2.4 NUMERICAL RESULTS IN THE FRAME OF METHOD OF SLOW COMPLEX
AMPLITUDES 129
6.3 PARAMETRIC GENERATOR OF HYPERBOLIC CHAOS BASED ON FOUR COUPLED
OSCILLATORS WITH PUMP MODULATION 131
6.3.1 MODEL, OPERATION PRINCIPLE AND BASIC EQUATIONS 132 6.3.2 CHAOTIC
DYNAMICS: RESULTS OF COMPUTER SIMULATION 134 REFERENCES 139
7 RECOGNIZING THE HYPERBOLIDTY: CONE CRITERION AND OTHER APPROACHES 141
7.1 VERIFICATION OF TRANSVERSALITY FOR MANIFOLDS 141
7.1.1 VISUALIZATION OF THE MANIFOLDS 142
7.1.2 DISTRIBUTIONS OF ANGLES OF THE MANIFOLD INTERSECTIONS 144 7.2
VISUALIZATION OF INVARIANT MEASURES 150
7.3 CONE CRITERION AND EXAMPLES OF ITS APPLICATION 155
7.3.1 PROCEDURE OF VERIFICATION OF THE CONE CRITERION 155 7.3.2 EXAMPLES
OF APPLICATION OF THE CONE CRITERION 161 REFERENCES 169
PART III HIGHER-DIMENSIONAL SYSTEMS AND PHENOMENA
8 SYSTEMS OF FOUR ALTERNATELY EXCITED NON-AUTONOMOUS OSCILLATORS . .
173 8.1 ARNOLD'S CAT MAP DYNAMICS IN A SYSTEM OF COUPLED NON-AUTONOMOUS
VAN DER POL OSCILLATORS 173
8.2 DYNAMICS CORRESPONDING TO HYPERCHAOTIC MAPS 180
8.2.1 SYSTEM IMPLEMENTING TORAL HYPERCHAOTIC MAP 180 8.2.2 MODEL WITH
CASCADE TRANSFER OF EXCITATION UPWARD THE FREQUENCY SPECTRUM 182
8.3 HYPERCHAOS AND SYNCHRONOUS CHAOS IN A SYSTEM OF COUPLED
NON-AUTONOMOUS OSCILLATORS 187
8.3.1 EQUATIONS AND BASIC MODES OF OPERATION 188
8.3.2 EQUATIONS FOR SLOW COMPLEX AMPLITUDES 193
REFERENCES 199
IMAGE 4
JYY CONTENTS
9 AUTONOMOUS SYSTEMS BASED ON DYNAMICS CLOSE TO HETERODINIC
CYCLE 201
9.1 HETEROCLINIC CONNECTION: AN EXAMPLE OF GUCKENHEIMER AND HOLMES 201
9.2 ATTRACTOR OF SMALE-WILLIAMS TYPE IN A SYSTEM OF THREE COUPLED
SELF-OSCILLATORS 203
9.3 ATTRACTOR WITH DYNAMICS GOVERNED BY THE ARNOLD CAT MAP 207 9.4 MODEL
WITH HYPERCHAOS 210
9.5 AN AUTONOMOUS SYSTEM WITH ATTRACTOR OF SMALE-WILLIAMS TYPE WITH
RESONANCE TRANSFER OF EXCITATION IN A RING ARRAY OF VAN DER POL
OSCILLATORS 213
REFERENCES 217
10 SYSTEMS WITH TIME-DELAY FEEDBACK 219
10.1 SOME NOTIONS CONCERNING DIFFERENTIAL EQUATIONS WITH DEVIATING
ARGUMENT 220
10.2 VAN DER POL OSCILLATOR WITH DELAYED FEEDBACK, PARAMETER MODULATION
AND AUXILIARY SIGNAL 223
10.2.1 ATTRACTOR OF SMALE-WILLIAMS TYPE IN THE TIME-DELAYED SYSTEM 224
10.2.2 HYPERCHAOTIC ATTRACTORS 227
10.3 VAN DER POL OSCILLATOR WITH TWO DELAYED FEEDBACK LOOPS AND
PARAMETER MODULATION 231
10.4 AUTONOMOUS TIME-DELAY SYSTEM 237
REFERENCES 240
11 CHAOS IN CO-OPERATIVE DYNAMICS OF ALTERNATELY SYNCHRONIZED ENSEMBLES
OF GLOBALLY COUPLED SELF-OSCILLATORS 243
11.1 KURAMOTO TRANSITION IN ENSEMBLE OF GLOBALLY COUPLED OSCILLATORS .
. 243 11.2 MODEL OF TWO ALTERNATELY SYNCHRONIZED ENSEMBLES OF
OSCILLATORS . . . 247 11.2.1 COLLECTIVE CHAOS IN ENSEMBLE OF VAN DER
POL OSCILLATORS 248 11.2.2 SLOW-AMPLITUDE APPROACH 251
11.2.3 DESCRIPTION OF THE DYNAMICS IN TERMS OF ENSEMBLES OF PHASE
OSCILLATORS 254
REFERENCES 256
PARTRV EXPERIMENTAL STUDIES
12 ELECTRONIC DEVICE WITH ATTRACTOR OF SMALE-WILLIAMS TYPE 259 12.1
SCHEME OF THE DEVICE AND THE PRINCIPLE OF OPERATION 259 12.2
EXPERIMENTAL OBSERVATION OF THE SMALE-WILLIAMS ATTRACTOR 260 REFERENCES
263
IMAGE 5
CONTENTS XV
13 DELAY-TIME ELECTRONIC DEVICES GENERATING TRAINS OF OSCILLATIONS WITH
PHASES GOVERNED BY CHAOTIC MAPS 265
13.1 VAN DER POL OSCILLATOR WITH DELAYED FEEDBACK, PARAMETER MODULATION
AND AUXILIARY SIGNAL 265
13.2 VAN DER POL OSCILLATOR WITH TWO DELAYED FEEDBACK LOOPS AND
PARAMETER MODULATION 269
REFERENCES 272
14 CONCLUSION 273
REFERENCES 275
APPENDIX A COMPUTATION OF LYAPUNOV EXPONENTS: THE BENETTIN ALGORITHM 277
REFERENCES 279
APPENDIX B HENON AND IKEDA MAPS 281
REFERENCES 287
APPENDIX C SMALE'S HORSESHOE AND HOMOCLINIC TANGLE 289 REFERENCES 292
APPENDIX D FRACTAL DIMENSIONS AND KAPLAN-YORKE FORMULA 293 REFERENCES
297
APPENDIX E HUNT'S MODEL: FORMAL DEFINITION 299
REFERENCES 303
APPENDIX F GEODESIES ON A COMPACT SURFACE OF NEGATIVE CURVATURE 305
REFERENCES 309
APPENDIX G EFFECT OF NOISE IN A SYSTEM WITH A HYPERBOLIC ATTRACTOR 311
REFERENCES 317
INDEX 319 |
| any_adam_object | 1 |
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| dewey-ones | 530 - Physics |
| dewey-raw | 530.15539 |
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| indexdate | 2024-09-06T00:22:12Z |
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| spelling | Kuznecov, Sergej P. Verfasser (DE-588)140545891 aut Hyperbolic chaos a physicist's view Sergey P. Kuznetsov Beijing Higher Education Press 2012 Berlin Springer XV, 320 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturangaben Attraktor (DE-588)4140563-8 gnd rswk-swf Nichtlineare Dynamik (DE-588)4126141-0 gnd rswk-swf Chaotisches System (DE-588)4316104-2 gnd rswk-swf Hyperbolizität (DE-588)4710615-3 gnd rswk-swf Nichtlineare Dynamik (DE-588)4126141-0 s Chaotisches System (DE-588)4316104-2 s Attraktor (DE-588)4140563-8 s Hyperbolizität (DE-588)4710615-3 s DE-604 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3863665&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=024780899&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kuznecov, Sergej P. Hyperbolic chaos a physicist's view Attraktor (DE-588)4140563-8 gnd Nichtlineare Dynamik (DE-588)4126141-0 gnd Chaotisches System (DE-588)4316104-2 gnd Hyperbolizität (DE-588)4710615-3 gnd |
| subject_GND | (DE-588)4140563-8 (DE-588)4126141-0 (DE-588)4316104-2 (DE-588)4710615-3 |
| title | Hyperbolic chaos a physicist's view |
| title_auth | Hyperbolic chaos a physicist's view |
| title_exact_search | Hyperbolic chaos a physicist's view |
| title_full | Hyperbolic chaos a physicist's view Sergey P. Kuznetsov |
| title_fullStr | Hyperbolic chaos a physicist's view Sergey P. Kuznetsov |
| title_full_unstemmed | Hyperbolic chaos a physicist's view Sergey P. Kuznetsov |
| title_short | Hyperbolic chaos |
| title_sort | hyperbolic chaos a physicist s view |
| title_sub | a physicist's view |
| topic | Attraktor (DE-588)4140563-8 gnd Nichtlineare Dynamik (DE-588)4126141-0 gnd Chaotisches System (DE-588)4316104-2 gnd Hyperbolizität (DE-588)4710615-3 gnd |
| topic_facet | Attraktor Nichtlineare Dynamik Chaotisches System Hyperbolizität |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3863665&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024780899&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kuznecovsergejp hyperbolicchaosaphysicistsview |