The physics of chaos in Hamiltonian systems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
London
Imperial College Press
2007
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XIV, 315 S. zahlr. Ill., graph. Darst. |
| ISBN: | 9781860947957 1860947956 |
Internformat
MARC
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| 245 | 1 | 0 | |a The physics of chaos in Hamiltonian systems |c George M. Zaslavsky |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a London |b Imperial College Press |c 2007 | |
| 300 | |a XIV, 315 S. |b zahlr. Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Chaotic behavior in systems | |
| 650 | 4 | |a Hamiltonian systems | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 0 | 7 | |a Chaostheorie |0 (DE-588)4009754-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hamiltonsches System |0 (DE-588)4139943-2 |2 gnd |9 rswk-swf |
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| 689 | 1 | 0 | |a Hamiltonsches System |0 (DE-588)4139943-2 |D s |
| 689 | 1 | 1 | |a Chaos |0 (DE-588)4191419-3 |D s |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-017556425 | ||
Datensatz im Suchindex
| _version_ | 1804139118794375168 |
|---|---|
| adam_text | CONTENTS
PREFACE TO THE FIRST EDITION
v
PREFACE TO THE SECOND EDITION
ix
1
DISCRETE AND CONTINUOUS MODELS
1
1.1
Coexistence of the Dynamical Order and Chaos
1
1.2
The Standard Map (Kicked Rotator)
3
1.3
The
Web
-Мар
(Kicked Oscillator)
8
1.4
Perturbed Pendulum
14
1.5
Perturbed Oscillator
17
1.6
Billiards
19
Conclusions
21
2
SEPARATRIX CHAOS
23
2.1
Nonlinear Resonance and Chain of Islands
23
2.2
Overlapping of Resonances
28
2.3
The Separatrix Map
30
2.4
Stochastic Layer
35
2.5
Hidden
Renormalisation
Group Near the Separatrix
41
2.6
Renormalisation
of Resonances
50
2.7
Stochastic Layer of the Standard Map
51
Conclusions
54
3
THE PHASE SPACE OF CHAOS
56
3.1
Non-universality of the Scenario
56
3.2
Collapsing Islands
63
xii
Contents
3.3
Blinking Islands 69
3.4
Boundary Islands
74
3.5
Self-Similar Set of Islands
76
*3.6 Ballistic Mode Islands
86
*3.7 General Comments About the Islands
88
Conclusions
89
4
NONLINEARITY VERSUS PERTURBATION
91
4.1
Beyond the KAM-Theory
91
4.2
Web-Tori
93
4.3
Width of the Stochastic Web
101
4.4
Transition from the KAM-Tori to Web-Tori
103
Conclusions
106
5
FRACTALS AND CHAOS
108
5.1
Fractal Dynamics
108
5.2
Generalised Fractal Dimension HI
5.3
Renormalisation
Group and Generalised Fractal
Dimension
113
5.4
Multi-Fractal Spectra
114
Conclusions
119
6
POINCARÉ
RECURRENCES AND FRACTAL
TIME
120
6.1
Poincaré
Recurrences
120
6.2
Poissonian Distribution of Recurrences
122
6.3
Non-Ergodicity, Stickiness and Quasi-Traps
125
6.4
Renormalisation Formulas for the Exit Time
Distribution
131
6.5
Fractal Time
135
6.6
Fractal and Multi-Fractal Recurrences
137
6.7
Multi-Fractal Space-Time and Its Dimension Spectrum
141
6.8
Critical Exponent for the
Poincaré
Recurrences
144
*6.9 Rhombic Billiard
145
Conclusions
148
Contents xiii
7 CHAOS AND FOUNDATION
OF STATISTICAL
PHYSICS
150
7.1
The Dynamical Foundation of Statistical Physics
150
7.2
Fractal Traps and Maxwell s Demon
152
7.3
Coupled Billiards
158
7.4
Contacted
Cassini-Sinai
Billiards
165
*7.5 Weak Mixing and Stickiness
172
*7.6 Persistent Fluctuations
173
Conclusions
176
8
CHAOS AND SYMMETRY
178
8.1
Stochastic Webs
178
8.2
Stochastic Web with Quasi-Crystalline Symmetry
181
8.3
Stochastic Web Skeleton
185
8.4
Symmetries and Their Dynamical Generation
195
8.5
The Width of the Stochastic Web
199
Conclusions
204
COLOUR PLATES (C.1-C.8)
9
MORE DEGREES OF FREEDOM
205
9.1
General Remarks
205
9.2
Four-Dimensional Map for the Motion in Magnetic
Field
206
9.3
Multi-Web
Structures in the Phase Space
210
9.4
Equilibrium of the Atomic Chains
215
9.5
Discretisation
219
Conclusions
222
10
NORMAL AND ANOMALOUS KINETICS
224
10.1 Fokker-Planck-Kolmogorov (FPK)
Equation
224
10.2
Transport for the Standard Map and Web-Map
229
10.3
Dynamics in the Potential with q-Fold Symmetry
233
10.4
More Examples of the Anomalous Transport
236
10.5
Levy Processes
239
10.6
The
Weierstrass
Random Walk
242
Conclusions
244
xiv Contents
11
FRACTIONAL KINETICS
245
11.1
Fractional Generalisation of the
Fokker-Planck-Kolmogorov Equation (FFPK)
245
11.2
Evolution of Moments
250
11.3
Method of the
Renormalisation
Group for Kinetics
(RGK)
252
*11.4 Complex Exponents and Log-Periodicity
257
Conclusions
260
*12 WEAK CHAOS AND
PSEUDOCHAOS
262
12.1
Definitions
262
12.2
Billiards with Pseudochaotic Dynamics
264
12.3
Filamented Surfaces
271
12.4
Bar-in-Square Billiard
273
12.5
Renormalisation
Group Equation for Recurrences
276
12.6
Recurrences in the Multi-Bar-Billiard
279
Conclusions
281
Appendix
1
THE NONLINEAR PENDULUM
283
Appendix
2
SOLUTION TO THE
RENORMALISATION
TRANSFORM EQUATION
287
*Appendix
3
FRACTIONAL INTEGRO-DIFFERENTIATION
289
*Appendix
4
FORMULAS OF FRACTIONAL CALCULUS
295
REFERENCES
300
INDEX
312
This book aims to familiarize the reader with the essential properties of the chaotic
dynamics of Hamiltonian systems by avoiding specialized mathematical tools, thus
making it easily accessible to a broader audience of researchers and students. Unique
material on the most intriguing and fascinating topics of unsolved and current problems
in contemporary chaos theory is presented. The coverage includes: separatrix chaos;
properties and a description of systems with non-ergodic dynamics; the distribution of
Poincaré
recurrences and their role in transport theory; dynamical models of the
Maxwell s Demon, the occurrence of persistent fluctuations, and a detailed discussion
of their role in the problem underlying the foundation of statistical physics; the emergence
of stochastic webs in phase space and their link to space tiling with periodic (crystal
type) and aperiodic (quasi-crystal type) symmetries.
This second edition expands on pseudochaotic dynamics with weak mixing and the
new phenomenon of fractional kinetics, which is crucial to the transport properties of
chaotic motion.
The book is ideally suited to all those who are actively working on the problems of
dynamical chaos as well as to those looking for new inspiration in this area. It introduces
the physicist to the world of Hamiltonian chaos and the mathematician to actual physical
problems.The material can also be used by graduate students.
|
| any_adam_object | 1 |
| author | Zaslavsky, George M. 1935-2008 |
| author_GND | (DE-588)13657985X |
| author_facet | Zaslavsky, George M. 1935-2008 |
| author_role | aut |
| author_sort | Zaslavsky, George M. 1935-2008 |
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| building | Verbundindex |
| bvnumber | BV035500187 |
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| callnumber-label | QC20 |
| callnumber-raw | QC20.7.H35 |
| callnumber-search | QC20.7.H35 |
| callnumber-sort | QC 220.7 H35 |
| callnumber-subject | QC - Physics |
| classification_rvk | UG 3900 |
| ctrlnum | (OCoLC)153579741 (DE-599)BVBBV035500187 |
| 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 239 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV035500187 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:39:00Z |
| institution | BVB |
| isbn | 9781860947957 1860947956 |
| language | English |
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| publisher | Imperial College Press |
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| spelling | Zaslavsky, George M. 1935-2008 Verfasser (DE-588)13657985X aut The physics of chaos in Hamiltonian systems George M. Zaslavsky 2. ed. London Imperial College Press 2007 XIV, 315 S. zahlr. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematische Physik Chaotic behavior in systems Hamiltonian systems Mathematical physics Chaostheorie (DE-588)4009754-7 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Chaos (DE-588)4191419-3 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 s Chaostheorie (DE-588)4009754-7 s DE-604 Chaos (DE-588)4191419-3 s Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017556425&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017556425&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Zaslavsky, George M. 1935-2008 The physics of chaos in Hamiltonian systems Mathematische Physik Chaotic behavior in systems Hamiltonian systems Mathematical physics Chaostheorie (DE-588)4009754-7 gnd Hamiltonsches System (DE-588)4139943-2 gnd Chaos (DE-588)4191419-3 gnd |
| subject_GND | (DE-588)4009754-7 (DE-588)4139943-2 (DE-588)4191419-3 |
| title | The physics of chaos in Hamiltonian systems |
| title_auth | The physics of chaos in Hamiltonian systems |
| title_exact_search | The physics of chaos in Hamiltonian systems |
| title_full | The physics of chaos in Hamiltonian systems George M. Zaslavsky |
| title_fullStr | The physics of chaos in Hamiltonian systems George M. Zaslavsky |
| title_full_unstemmed | The physics of chaos in Hamiltonian systems George M. Zaslavsky |
| title_short | The physics of chaos in Hamiltonian systems |
| title_sort | the physics of chaos in hamiltonian systems |
| topic | Mathematische Physik Chaotic behavior in systems Hamiltonian systems Mathematical physics Chaostheorie (DE-588)4009754-7 gnd Hamiltonsches System (DE-588)4139943-2 gnd Chaos (DE-588)4191419-3 gnd |
| topic_facet | Mathematische Physik Chaotic behavior in systems Hamiltonian systems Mathematical physics Chaostheorie Hamiltonsches System Chaos |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017556425&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017556425&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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