Chaos:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hackensack, NJ [u.a.]
World Scientific
2010
|
| Schriftenreihe: | Series on advances in statistical mechanics
17 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XX, 460 S. Ill., graph. Darst. |
| ISBN: | 9789814277655 |
Internformat
MARC
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| 100 | 1 | |a Cencini, Massimo |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Chaos |c Massimo Cencini |
| 264 | 1 | |a Hackensack, NJ [u.a.] |b World Scientific |c 2010 | |
| 300 | |a XX, 460 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Series on advances in statistical mechanics |v 17 | |
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Datensatz im Suchindex
| _version_ | 1804143567407415296 |
|---|---|
| adam_text | Contents
Preface
v
Introduction
vii
Introduction to Dynamical Systems and Chaos
1.
First Encounter with Chaos
3
1.1
Prologue
................................. 3
1.2
The nonlinear pendulum
........................ 3
1.3
The damped nonlinear pendulum
. ■.................. 5
1.4
The vertically driven and damped nonlinear pendulum
....... 6
1.5
What about the predictability of pendulum evolution?
....... 8
1.6
Epilogue
................................. 10
2.
The Language of Dynamical Systems
11
2.1
Ordinary Differential Equations (ODE)
............... 11
2.1.1
Conservative and dissipative dynamical systems
...... 13
Box B.I Hamiltonian dynamics
.................... 15
2.1.2
PoincaréMap
......................... 19
2.2
Discrete time dynamical systems: maps
............... 20
2.2.1
Two dimensional maps
.................... 21
2.3
The role of dimension
......................... 25
2.4
Stability theory
............................. 26
2.4.1
Classification of fixed points and linear stability analysis
. 27
BoxB.2 A remark on the linear stability of symplectic maps
.... 29
2.4.2
Nonlinear stability
....................... 30
2.5
Exercises
................................ 33
3.
Examples of Chaotic Behaviors
37
3.1
The logistic map
............................ 37
xvi
Chaos:
From Simple Models to Complex Systems
BoxB.3
Topologicul conjugacy
..................... 45
3.2
The
Lorenz
model
........................... 46
BoxB.4 Derivation of the
Lorenz
model
............... 51
3.3
The
Hénon-Heiles
system
....................... 53
3.4
What did we learn and what will we learn?
............. 58
BoxB.5 Correlation functions
..................... 61
3.5
Closing remark
............................. 62
3.6
Exercises
................................ 62
4.
Probabilistic Approach to Chaos
65
4.1
An informal probabilistic approach
.................. 65
4.2
Time evolution of the probability density
.............. 68
BoxB.6 Markov Processes
....................... 72
4.3
Ergodicity
................................ 77
4.3.1
An historical interlude on ergodic theory
.......... 78
BoxB.7
Poincaré
recurrence theorem
................. 79
4.3.2
Abstract formulation of the Ergodic theory
......... 81
4.4
Mixing
.................................. 84
4.5
Markov chains and chaotic maps
................... 86
4.6
Natural measure
............................ 89
4.7
Exercises
................................ 91
5.
Characterization of Chaotic Dynamical Systems
93
5.1
Strange attractors
........................... 93
5.2
Fractals and multifractals
....................... 95
5.2.1
Box counting dimension
.................... 98
5.2.2
The stretching and folding mechanism
............ 100
5.2.3
Multifractals
.......................... 103
BoxB.8 Brief excursion on Large Deviation Theory
........ 108
5.2.4
Grassberger-Procaccia algorithm
............... 109
5.3
Characteristic Lyapunov exponents
..................
Ill
BoxB.9 Algorithm for computing Lyapunov Spectrum
....... 115
5.3.1
Oseledec theorem and the law of large numbers
...... 116
5.3.2
Remarks on the Lyapunov exponents
............ 118
5.3.3
Fluctuation statistics of finite time Lyapunov exponents
. 120
5.3.4
Lyapunov dimension
..................... 123
Box B.
10
Mathematical chaos
..................... 124
5.4
Exercises
................................ 127
6.
From Order to Chaos in Dissipative Systems
131
6.1
The scenarios for the transition to turbulence
............ 131
6.1.1 Landau-Hopf.......................... 132
Box
В.
11 Hopf
bifurcation
....................... 134
Contents xvii
BoxB.12
The Van
der Pol
oscillator and the averaging technique
. 135
6.1.2
Ruelle-Takens
......................... 137
6.2
The period doubling transition
.................... 139
6.2.1 Feigenbaum renormalization
group
.............. 142
6.3
Transition to chaos through intermittency: Pomeau-Manneville
scenario
................................. 145
6.4
A mathematical remark
........................ 147
6.5
Transition to turbulence in real systems
............... 148
6.5.1
A visit to laboratory
..................... 149
6.6
Exercises
................................ 151
7.
Chaos in Hamiltonian Systems
153
7.1
The integrability problem
....................... 153
7.1.1
Poincaré
and the non-existence of integrals of motion
... 154
7.2
Ko
Imogorov-Arnold-
Moser
theorem and the survival of tori
.... 155
BoxB.13 Arnold diffusion
....................... 160
7.3
Poincaré-Birkhoff
theorem and the fate of resonant tori
...... 161
7.4
Chaos around
séparatrices
...................... 164
BoxB.14 The resonance-overlap criterion
.............. 168
7.5
Melnikov s theory
........................... 171
7.5.1
An application to the Duffing s equation
.......... 174
7.6
Exercises
................................ 175
Advanced Topics and Applications: From Information
Theory to Turbulence
8.
Chaos and Information Theory
179
8.1
Chaos, randomness and information
.................179
8.2
Information theory, coding and compression
.............183
8.2.1
Information sources
...................... 184
8.2.2
Properties and uniqueness of entropy
............ 185
8.2.3
Shannon entropy rate and its meaning
........... 187
BoxB.15 Transient behavior of block-entropies
........... 190
8.2.4
Coding and compression
................... 192
8.3
Algorithmic complexity
........................ 194
Box B.I
6
Ziv-Lempel compression algorithm
............. 196
8.4
Entropy and complexity in chaotic systems
............. 197
8.4.1
Partitions and symbolic dynamics
.............. 197
8.4.2
Kolmogorov-Sinai entropy
.................. 200
BoxB.
17
Rényi
entropies
....................... 203
8.4.3
Chaos, unpredictability and uncompressibility
....... 203
xviii
Chaos: Fmm
Simple Models
to Complex Systems
8.5
Concluding remarks
..........................205
8.6
Exercises
................................206
9.
Coarse-Grained Information and Large Scale Predictability
209
9.1
Finite-resolution versus infinite-resolution descriptions
.......209
9.2
ε
-entropy in information theory: lossless versus lossy coding
.... 213
9.2.1
Channel capacity
....................... 213
9.2.2
Rate distortion theory
..................... 215
Box B.
18
ε
-entropy for the Bernoulli and Gaussian source
..... 218
9.3
ε
-entropy in dynamical systems and stochastic processes
..... 219
9.3.1
Systems classification according to
ε
-entropy behavior
. . . 222
Box B.
19
ε
-entropy from exit-times statistics
............ 224
9.4
The finite size lyapunov exponent (FSLE)
.............. 228
9.4.1
Linear vs nonlinear instabilities
...............233
9.4.2
Predictability in systems with different characteristic times
234
9.5
Exercises
................................237
10.
Chaos in Numerical and Laboratory Experiments
239
10.1
Chaos in silico
.............................239
Box B.
20
Round-off errors and floating-point representation
.... 241
10.1.1
Shadowing lemma
....................... 242
10.1.2
The effects of state discretization
.............. 244
Box B.
21
Effect of discretization: a probabilistic argument
..... 247
10.2
Chaos detection in experiments
.................... 247
Box B.
22
Lyapunov exponents from experimental data
....... 250
10.2.1
Practical difficulties
...................... 251
10.3
Can chaos be distinguished from noise?
............... 255
10.3.1
The finite resolution analysis
................. 256
10.3.2
Scale-dependent signal classification
............. 256
10.3.3
Chaos or noise? A puzzling dilemma
............ 258
10.4
Prediction and modeling from data
.................. 263
10.4.1
Data prediction
........................263
10.4.2
Data modeling
.........................264
11.
Chaos in Low Dimensional Systems
267
11.1
Celestial mechanics
...........................267
11.1.1
The restricted three-body problem
.............. 269
11.1.2
Chaos in the Solar system
.................. 273
Box B.23 A symplectic map for
Halley
comet
............ 276
11.2
Chaos and transport phenomena in fluids
.............. 279
Box B.24 Chaos and passive scalar transport
............ 280
11.2.1
Lagrangian chaos
....................... 283
Contents xix
Box
В.
25
Point vortices and the two-dimensional
Euler
equation
. 288
11.2.2
Chaos and diffusion in laminar flows
............. 290
Box B.
26
Relative dispersion in turbulence
.............. 295
11.2.3
Advection of
inerţial
particles
................ 296
11.3
Chaos in population biology and chemistry
.............299
11.3.1
Population biology:
Lotka-
Volterra systems
......... 300
11.3.2
Chaos in generalized
Lotka-
Volterra systems
........ 304
11.3.3
Kinetics of chemical reactions: Belousov-Zhabotinsky
. . . 307
BoxB.27 Michaelis-Menten law of simple enzymatic reaction
. . . 311
11.3.4
Chemical clocks
........................ 312
Box B.
28
A model for biochemical oscillations
............ 314
11.4
Synchronization of chaotic systems
.................. 316
11.4.1
Synchronization of regular oscillators
............317
11.4.2
Phase synchronization of chaotic oscillators
.........319
11.4.3
Complete synchronization of chaotic systems
........323
12.
Spatiotemporal
Chaos
329
12.1
Systems and models for
spatiotemporal
chaos
............329
12.1.1
Overview of
spatiotemporal
chaotic systems
........ 330
12.1.2
Networks of chaotic systems
................. 337
12.2
The thermodynamic limit
....................... 338
12.3
Growth and propagation of space-time perturbations
........ 340
12.3.1
An overview
.......................... 340
12.3.2
Spatial and Temporal Lyapunov exponents
...... 341
12.3.3
The comoving Lyapunov exponent
.............. 343
12.3.4
Propagation of perturbations
................. 344
Box B.
29
Stable chaos and supertransients
.............. 348
12.3.5
Convective chaos and sensitivity to boundary conditions
. 350
12.4
Non-equilibrium phenomena and
spatiotemporal
chaos
....... 352
Box B.
30
Non-equilibrium phase transitions
............. 353
12.4.1
Spatiotemporal
perturbations and interfaces roughening
. 356
12.4.2
Synchronization of extended chaotic systems
........ 358
12.4.3
Spatiotemporal intermittency
................ 361
12.5
Coarse-grained description of high dimensional chaos
........ 363
12.5.1
Scale-dependent description of high-dimensional systems
. 363
12.5.2
Macroscopic chaos: low dimensional dynamics embedded
in high dimensional chaos
..................365
13.
Turbulence as a Dynamical System Problem
369
13.1
Fluids as dynamical systems
......................369
13.2
Statistical mechanics of ideal fluids and turbulence phenomenology
373
13.2.1
Three dimensional ideal fluids
................373
xx
Chaos:
Prom
Simple Models
to Complex
Systems
13.2.2
Two
dimensional ideal
fluids.................
374
13.2.3
Phenomenology of three
dimensional
turbulence
...... 375
Box
В.
31
Intermittency in three-dimensional turbulence:
the multifractal model
..................... 379
13.2.4
Phenomenology of two dimensional turbulence
....... 382
13.3
From partial differential equations to ordinary differential
equations
................................ 385
13.3.1
On the number of degrees of freedom of turbulence
.... 385
13.3.2
The Galerkin method
..................... 387
13.3.3
Point vortices method
..................... 388
13.3.4
Proper
orthonormal
decomposition
............. 390
13.3.5
Shell models
.......................... 391
13.4
Predictability in turbulent systems
.................. 394
13.4.1
Small scales predictability
.................. 395
13.4.2
Large scales predictability
.................. 397
13.4.3
Predictability in the presence of coherent structures
. . . 401
14.
Chaos and Statistical Mechanics: Fermi-Pasta-
Ulam
a Case Study
405
14.1
An influential unpublished paper
................... 405
14.1.1
Toward an explanation:
Solitons
or
KAM?......... 409
14.2
A random walk on the role of ergodicity and chaos for equilibrium
statistical mechanics
.......................... 411
14.2.1
Beyond metrical transitivity: a physical point of view
... 411
14.2.2
Physical questions and numerical results
.......... 412
14.2.3
Is chaos necessary or sufficient for the validity of statistical
mechanical laws?
....................... 415
14.3
Final remarks
.............................. 417
Box B.
32
Pseudochaos
and diffusion
................. 418
Epilogue
421
Bibliography
427
Index
455
|
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| id | DE-604.BV036865349 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:49:43Z |
| institution | BVB |
| isbn | 9789814277655 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020781032 |
| oclc_num | 699053515 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR |
| owner_facet | DE-355 DE-BY-UBR |
| physical | XX, 460 S. Ill., graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | World Scientific |
| record_format | marc |
| series | Series on advances in statistical mechanics |
| series2 | Series on advances in statistical mechanics |
| spelling | Cencini, Massimo Verfasser aut Chaos Massimo Cencini Hackensack, NJ [u.a.] World Scientific 2010 XX, 460 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Series on advances in statistical mechanics 17 Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Chaostheorie (DE-588)4009754-7 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 s Chaostheorie (DE-588)4009754-7 s DE-604 Series on advances in statistical mechanics 17 (DE-604)BV000019119 17 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020781032&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cencini, Massimo Chaos Series on advances in statistical mechanics Funktionentheorie (DE-588)4018935-1 gnd Chaostheorie (DE-588)4009754-7 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4009754-7 |
| title | Chaos |
| title_auth | Chaos |
| title_exact_search | Chaos |
| title_full | Chaos Massimo Cencini |
| title_fullStr | Chaos Massimo Cencini |
| title_full_unstemmed | Chaos Massimo Cencini |
| title_short | Chaos |
| title_sort | chaos |
| topic | Funktionentheorie (DE-588)4018935-1 gnd Chaostheorie (DE-588)4009754-7 gnd |
| topic_facet | Funktionentheorie Chaostheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020781032&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000019119 |
| work_keys_str_mv | AT cencinimassimo chaos |