Verfügbarkeit: Hamiltonian chaos and fractional dynamics

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Titel:	Hamiltonian chaos and fractional dynamics
Von:	George M. Zaslavsky
Person:	Zaslavsky, George M. 1935-2008 Verfasser aut
Hauptverfasser:	Zaslavsky, George M. 1935-2008 (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Oxford [u.a.] Oxford Univ. Press 2006
Ausgabe:	Reprinted
Notation:	SK 950
Schlagwörter:	aHamiltonian systems aChaotic behavior in systems Hamilton-Formalismus Nichtlineares dynamisches System Chaotisches System Hamiltonsches System
Medienzugang:	http://www.loc.gov/catdir/toc/ecip0421/2004018403.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013083272&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Beschreibung:	Includes bibliographical references and index
Umfang:	XIV, 421 S. Ill., graph. Darst.
ISBN:	0198526040

Internformat

MARC


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Datensatz im Suchindex

DE-BY-UBR_call_number	84/SK 950 Z38
DE-BY-UBR_katkey	4343060
DE-BY-UBR_location	UB Lesesaal Physik
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adam_text	CONTENTS Part 1 Chaotic Dynamics 1 Hamiltonian dynamics 3 1.1 Hamiltonian equations 3 1.2 Phase space dynamics 5 1.3 Action-angle variable (one degree of freedom) 8 Notes Problems 2 Examples of Hamiltonian dynamics 13 2.1 Pendulum 13 2.2 Oscillations in the infinite potential well 16 2.3 Magnetic moments 17 2.4 Field line behaviour 18 2.5 Hamiltonian equations for the ABC-flow 20 Problems 22 3 Perturbed dynamics 23 3.1 The Liouville-Arnold theorem on integrability 23 3.2 Consequences of the integrability 25 3.3 Non-integrability and the Kozlov condition 26 3.4 Resonances 28 3.5 Non-linear resonance and chain of islands 29 3.6 Kolmogorov-Arnold-Moser (KAM) theory 32 Notes Problems 4 Chaotic dynamics 37 4.1 Natural measure 37 4.2 Ergodicity, mixing, and weak mixing 39 4.3 Local instability and Lýapunov exponents 42 4.4 Hyperbolic systems 46 4.5 Entropy of dynamical systems 48 4.5.1 Partitioning and coarse-graining 48 4.5.2 Kolmogorov-Smai (KS) entropy 49 4.5.3 Topological entropy 51 4.5.4 Physical interpretation 51 4.5.5 Entropy and Lyapunov exponents 53 CONTENTS 4.6 Definition of chaotic dynamics 53 4.7 Chirikov resonance overlapping criteria 54 Notes Problems Physical models of chaos 57 5.1 Mapping the dynamics 57 5.2 Universal and standard map 60 5.3 Web map (kicked oscillator) 64 5.4 Kepler map 68 Notes Problems Separatrix chaos 73 6.1 Description of models 73 6.2 Separatrix map 76 6.3 The stochastic layer 78 6.4 The stochastic layer of the standard map 81 6.5 Hidden renormalization group near the separatrix 83 6.6 Renormalization of resonances 89 6.7 Hidden renormalization for coupled oscillators 91 Notes Problems Weak chaos and symmetry 97 7.1 Stochastic webs 97 7.2 Stochastic webs with quasi-crystalline symmetry 99 7.3 Stochastic web skeleton 102 7.4 Symmetries and their dynamical generation 110 7.5 Width of the stochastic web 114 7.6 Symmetry in art and nature 117 7.6.1 Symmetry and chaos 117 7.6.2 Ornamental patterns 118 7.6.3 Patterns in nature 120 Notes Problems Beyond the KAM-theory 125 8.1 Small non-linearity 125 8.2 Web-Tori 127 8.3 Width of the stochastic web 134 8.4 Transition from KAM-Tori to Web-Tori 135 Notes Problems CONTENTS 9 Phase space of chaos 139 9.1 Topological non- universality of chaos 139 9.2 Examples with billiards 142 9.3 Accelerator mode islands 143 9.4 Ballistic mode islands 151 9.5 Cantori 152 9.6 Sticky domains and escapes 154 Notes Problems Part 2 Practality of Chaos 10 Fractals and chaos 159 10.1 Fractal dynamics 159 10.2 Generalized fractal dimension 161 10.3 Renormalization group and generalized fractal dimension 162 10.4 Multifractal spectra 164 10.5 Thermodynamic interpretation 167 10.6 Complex dimension and log-periodicity 169 Notes Problems 11 Poincaré recurrences 173 11.1 Poincaré theorem on recurrences 173 11.2 Recurrence time distributions and Кас lemma 174 11.3 Distribution of recurrences in uniform mixing 177 11.4 More asymptotics on recurrences 180 Notes Problems 12 Dynamical traps 187 12.1 Definition of the dynamical trap 187 12.2 Hierarchical-islands trap (HIT) 189 12.3 Renormalization for the exit time distribution 193 12.4 Stochastic layer trap 196 Notes 13 Fractal time 201 13.1 Fractal time 201 13.2 Fractal and multifractal recurrences 204 13.3 Multifractal space-time and its dimension spectrum 207 13.4 Critical exponent for the Poincaré recurrences 209 Notes Problems CONTENTS Part 3 Chaotic kinetics 14 General principles of kinetics 215 14.1 Time scales 215 14.2 Fokker-Planck-Kolmogorov (FPK) equation 217 14.3 Detailed balance principle 220 14.4 Solutions and normal transport 221 14.5 Growth of entropy 222 14.6 Kolmogorov conditions and conflict with dynamics 223 14.7 Truncated distributions 225 Notes Problems 15 Levy process, Levy flights, and Weierstrass random walk 229 15.1 Levy distribution 230 15.2 Levy process 231 15.3 Poincaré recurrences and Feller s theorems 234 15.4 Levy flights and conflict with dynamics 235 15.5 Weirstrass random walks (WRW) 240 Notes Problems 16 Fractional kinetic equation (FKE) 245 16.1 Derivation of FKE 245 16.2 Conditions for the FKE 249 16.3 Evolution of moments (transport) 250 16.4 Conflict with dynamics 252 16.5 Dynamical origin of critical exponents 253 16.6 Principles of simulations 257 Notes Problems 17 Renormalization group of kinetics (RGK) 261 17.1 Space-time scalings 261 17.2 Log-periodicity 263 17.3 Duality of the dynamics and the origin of multi-fractality 265 17.4 Multifractional kinetics 267 Notes 18 Fractional kinetic equation: solutions and modifications 273 18.1 Solutions to FKE (series) 273 18.2 Solutions to FKE (separation of variables) 275 18.3 Continuous time random walk (CTRW) 276 18.4 Levy walks and other generalizations of CTRW 279 18.5 Conflict with dynamics 281 18.6 Subdiffusion and superdiffusion 281 CONTENTS Notes Problems 19 Pseudochaos 287 19.1 Billiards in polygons 287 19.2 Continued fractions and scalings of trajectories 291 19.3 Fractional kinetics of irrational trajectories 296 19.4 More examples of pseudochaos 303 19.4.1 Rhombic billiard 303 19.4.2 More billiards 303 19.4.3 Saw-tooth web map 306 Notes Problems Part 4 Applications 20 Complexity and entropy of dynamics 315 20.1 Complexity in phase space 316 20.2 Symbolic and topological complexities 317 20.3 Topological and metric entropies 320 20.4 Conflict with dynamics 323 Notes Problems 21 Complexity and entropy functions 325 21.1 Definitions of complexity function 325 21.2 Probability of e-divergence 328 21.3 Calculation of local complexity function 329 21.4 Flight complexity function 331 21.5 Entropy function 333 21.6 Polynomial and mixed complexities and anomalous transport 335 21.7 Travelling waves and Riemann invariants of entropy and complexity 337 Notes Problems 22 Chaos and foundation of statistical mechanics 341 22.1 Zermelo s and Loschmidt s paradoxes 341 22.1.1 Historical comments 341 22.1.2 Paradox of recurrence 342 22.1.3 Paradox of reversibility 343 22.1.4 Boltzmann s comments 343 22.2 Chaos and the paradoxes 344 22.3 Anomalous properties of the Sinai and Bunimovich billiards 344 22.4 Maxwell s Demon and Chaos 346 CONTENTS 22.5 Maxwell s Demon as a dynamical model 348 22.6 Comments on the application of Ergodic Theory 352 22.7 Comments on dynamical cooling and chaos erasing 352 Notes 23 Chaotic advection (dynamics of tracers) 357 23.1 Beltrami flows with g-symmetry 357 23.2 Compressible helical flows 359 23.3 Compressible flow with quasi-symmetry 367 Notes Problems 24 Advection by point vortices 373 24.1 Basic equations for point vortices and for advection 373 24.2 Advection in three vortices 376 24.3 Transport of advected particles (vortices) 383 Notes Problems Appendices 393 A Elliptic integrals and elliptic functions 393 В Spectrum of the Kepler problem 394 С Fractional integro-diferentiation 396 D Formulas of fractional calculus 399 References 403 Index 417
any_adam_object	1
author	Zaslavsky, George M. 1935-2008
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id	DE-604.BV019756825
illustrated	Illustrated
indexdate	2024-12-20T12:05:11Z
institution	BVB
isbn	0198526040
language	English
lccn	2004018403
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-013083272
oclc_num	255571210
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owner	DE-703 DE-355 DE-BY-UBR
owner_facet	DE-703 DE-355 DE-BY-UBR
physical	XIV, 421 S. Ill., graph. Darst.
publishDate	2006
publishDateSearch	2006
publishDateSort	2006
publisher	Oxford Univ. Press
record_format	marc
spellingShingle	Zaslavsky, George M. 1935-2008 Hamiltonian chaos and fractional dynamics aHamiltonian systems aChaotic behavior in systems Hamilton-Formalismus (DE-588)4376155-0 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd Chaotisches System (DE-588)4316104-2 gnd Hamiltonsches System (DE-588)4139943-2 gnd
subject_GND	(DE-588)4376155-0 (DE-588)4126142-2 (DE-588)4316104-2 (DE-588)4139943-2
title	Hamiltonian chaos and fractional dynamics
title_auth	Hamiltonian chaos and fractional dynamics
title_exact_search	Hamiltonian chaos and fractional dynamics
title_full	Hamiltonian chaos and fractional dynamics George M. Zaslavsky
title_fullStr	Hamiltonian chaos and fractional dynamics George M. Zaslavsky
title_full_unstemmed	Hamiltonian chaos and fractional dynamics George M. Zaslavsky
title_short	Hamiltonian chaos and fractional dynamics
title_sort	hamiltonian chaos and fractional dynamics
topic	aHamiltonian systems aChaotic behavior in systems Hamilton-Formalismus (DE-588)4376155-0 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd Chaotisches System (DE-588)4316104-2 gnd Hamiltonsches System (DE-588)4139943-2 gnd
topic_facet	aHamiltonian systems aChaotic behavior in systems Hamilton-Formalismus Nichtlineares dynamisches System Chaotisches System Hamiltonsches System
url	http://www.loc.gov/catdir/toc/ecip0421/2004018403.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013083272&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT zaslavskygeorgem hamiltonianchaosandfractionaldynamics

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