Lyapunov exponents: a tool to explore complex dynamics
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom
Cambridge University Press
2016
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Literaturverzeichnis Seite 259-276 |
| Beschreibung: | xii, 285 Seiten Porträt, Diagramme |
| ISBN: | 9781107030428 |
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Datensatz im Suchindex
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| adam_text | Preface page xi
1 Introduction 1
1.1 Historical considerations 1
1.1.1 Early results 1
1.1.2 Biography of Aleksandr Lyapunov 3
1.1.3 Lyapunov’s contribution 4
1.1.4 The recent past 5
1.2 Outline of the book 6
1.3 Notations 8
2 The basics 10
2.1 The mathematical setup 10
2.2 One-dimensional maps 11
2.3 Oseledets theorem 12
2.3.1 Remarks 13
2.3.2 Oseledets splitting 15
2.3.3 “Typical perturbations” and time inversion 16
2.4 Simple examples 17
2.4.1 Stability of fixed points and periodic orbits 17
2.4.2 Stability of independent and driven systems 18
2.5 General properties 18
2.5.1 Deterministic vs. stochastic systems 18
2.5.2 Relationship with instabilities and chaos 19
2.5.3 Invariance 20
2.5.4 Volume contraction 21
2.5.5 Time parametrisation 22
2.5.6 Symmetries and zero Lyapunov exponents 24
2.5.7 Symplectic systems 26
3 Numerical methods 28
3.1 The largest Lyapunov exponent 28
3.2 Full spectrum: QR decomposition 29
3.2.1 Gram-Schmidt orthogonalisation 31
3.2.2 Householder reflections 31
v
Vi
Contents
3.3 Continuous methods 33
3.4 Ensemble averages 35
3.5 Numerical errors 36
3.5.1 Orthogonalisation 37
3.5.2 Statistical error 38
3.5.3 Near degeneracies 40
3.6 Systems with discontinuities 43
3.6.1 Pulse-coupled oscillators 48
3.6.2 Colliding pendula 49
3.7 Lyapunov exponents from time series 50
4 Lyapunov vectors 54
4.1 Forward and backward Oseledets vectors 55
4.2 Covariant Lyapunov vectors and the dynamical algorithm 57
4.3 Dynamical algorithm: numerical implementation 59
4.4 Static algorithms 61
4.4.1 Wolfe-Samelson algorithm 62
4.4.2 Kuptsov-Parlitz algorithm 62
4.5 Vector orientation 63
4.6 Numerical examples 64
4.7 Further vectors 65
4.7.1 Bred vectors 66
4.7.2 Dual Lyapunov vectors 67
5 Fluctuations, finite-time and generalised exponents 70
5.1 Finite-time analysis 70
5.2 Generalised exponents 73
5.3 Gaussian approximation 77
5.4 Numerical methods 78
5.4.1 Quick tools 78
5.4.2 Weighted dynamics 79
5.5 Eigenvalues of evolution operators 80
5.6 Lyapunov exponents in terms of periodic orbits 84
5.7 Examples 89
5.7.1 Deviation from hyperbolicity 89
5.7.2 Weak chaos 90
5.7.3 Henon map 94
5.7.4 Mixed dynamics 96
6 Dimensions and dynamical entropies 100
6.1 Lyapunov exponents and fractal dimensions 100
6.2 Lyapunov exponents and escape rate 103
6.3 Dynamical entropies 105
Contents
6.4 Generalised dimensions and entropies 107
6.4.1 Generalised Kaplan-Yorke formula 107
6.4.2 Generalised Pesin formula 109
7 Finite-amplitude exponents 110
7.1 Finite vs. infinitesimal perturbations 110
7.2 Computational issues 112
7.2.1 One-dimensional maps 114
7.3 Applications 115
8 Random systems 118
8.1 Products of random matrices 119
8.1.1 Weak disorder 119
8.1.2 Highly symmetric matrices 125
8.1.3 Sparse matrices 128
8.1.4 Polytomic noise 131
8.2 Linear stochastic systems and stochastic stability 136
8.2.1 First-order stochastic model 136
8.2.2 Noise-driven oscillator 137
8.2.3 Khasminskii theory 141
8.2.4 High-dimensional systems 142
8.3 Noisy nonlinear systems 146
8.3.1 LEs as eigenvalues and supersymmetry 146
8.3.2 Weak-noise limit 149
8.3.3 Synchronisation by common noise and random attractors 150
9 Coupled systems 152
9.1 Coupling sensitivity 152
9.1.1 Statistical theory and qualitative arguments 15 3
9.1.2 Avoided crossing of LEs and spacing statistics 157
9.1.3 A statistical-mechanics example 159
9.1.4 The zero exponent 160
9.2 Synchronisation 162
9.2.1 Complete synchronisation and transverse Lyapunov exponents 162
9.2.2 Clusters, the evaporation and the conditional Lyapunov exponent 163
9.2.3 Synchronisation on networks and master stability function 164
10 High-dimensional systems: general 168
10.1 Lyapunov density spectrum 168
10.1.1 Infinite systems 171
10.2 Chronotopic approach and entropy potential 173
10.3 Convective exponents and propagation phenomena 178
10.3.1 Mean-field approach 181
Contents
10.3.2 Relationship between convective exponents and chronotopic
analysis 183
10.3.3 Damage spreading 185
10.4 Examples of high-dimensional systems 187
10.4.1 Hamiltonian systems 187
10.4.2 Differential-delay models 191
10.4.3 Long-range coupling 193
11 High-dimensional systems: Lyapunov vectors and finite-size effects 200
11.1 Lyapunov dynamics as a roughening process 200
11.1.1 Relationship with the KPZ equation 202
11.1.2 The bulk of the spectrum 207
11.2 Localisation of the Lyapunov vectors and coupling sensitivity 209
11.3 Macroscopic dynamics 213
11.3.1 From micro to macro 216
11.3.2 Hydrodynamic Lyapunov modes 218
11.4 Fluctuations of the Lyapunov exponents in space-time chaos 219
11.5 Open system approach 223
11.5.1 Lyapunov spectra of open systems 226
11.5.2 Scaling behaviour of the invariant measure 226
12 Applications 229
12.1 Anderson localisation 229
12.2 Billiards 231
12.3 Lyapunov exponents and transport coefficients 235
12.3.1 Escape rate 235
12.3.2 Molecular dynamics 236
12.4 Lagrangian coherent structures 237
12.5 Celestial mechanics 239
12.6 Quantum chaos 242
Appendix A Reference models 245
A. 1 Lumped systems: discrete time 245
A.2 Lumped systems: continuous time 246
A.3 Lattice systems: discrete time 247
A.4 Lattice systems: continuous time 248
A.5 Spatially continuous systems 249
A.6 Differential-delay systems 250
A.7 Global coupling: discrete time 250
A.8 Global coupling: continuous time 250
Appendix B Pseudocodes
252
IX
Contents
Appendix C Random matrices: some general formulas 256
C. 1 Gaussian matrices: discrete time 256
C,2 Gaussian matrices: continuous time 257
Appendix D Symbolic encoding 258
Bibliography 259
Index 211
LYAPUNOV EXPONENTS
L/apunov exponents lie at the heart of chaos theory and are widely used in studies of
complex dynamics. Utilising a pragmatic, physical approach, this self-contained book
provides a comprehensive description of the concept. Beginning with the basic properties
and numerical methods, it then guides readers through to the most recent advances
in applications to complex systems. Practical algorithms are thoroughly reviewed and
their performance is discussed, while a broad set of examples illustrate the wide range
of potential applications. The description of various numerical and analytical techniques
for the computation of Lyapunov exponents offers an extensive array of tools for the
characterisation of phenomena such as synchronisation, weak and global chaos in low
and high-dimensional setups, and localisation. This text equips readers with all the
investigative expertise needed to explore fully the dynamical properties of complex
systems, making it ideal for both graduate students and experienced researchers.
ARKADY PIKOVSKY is Professor of Theoretical Physics at the University of Potsdam.
He is a member of the editorial board for Physica D and a Chaotic and Complex Systems
Editor for J. Physics A: Mathematical and Theoretical. He is a Fellow of the American Physical
Society and co-author of Synchronization: A Universal Concept in Nonlinear Sciences. His
current research focusses on nonlinear physics of complex systems.
ANTONIO POLITI is the 6th Century Chair in Physics of Life Sciences at the
University of Aberdeen. He is Associate Editor of Physical Review E, a Fellow of the
Institute of Physics and of the American Physical Society and was awarded the Gutzwiller
Prize by the Max-Planck Institute for Complex Systems in Dresden and the Humboldt
Prize. He is co-author of Complexity: Hierarchical Structures and Scaling in Physics.
ISBN 978-1 -
9 781 1
07-03042-8
[3 )42Í■
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| spelling | Pikovskij, Arkadij 1956- Verfasser (DE-588)140665773 aut Lyapunov exponents a tool to explore complex dynamics Arkady Pikovsky, University of Potsdam, Antonio Politi, University of Aberdeen Cambridge, United Kingdom Cambridge University Press 2016 xii, 285 Seiten Porträt, Diagramme txt rdacontent n rdamedia nc rdacarrier Literaturverzeichnis Seite 259-276 Lyapunov exponents Differential equations Ljapunov-Exponent (DE-588)4123668-3 gnd rswk-swf Ljapunov-Exponent (DE-588)4123668-3 s DE-604 Politi, Antonio 1975- Verfasser (DE-588)132545829 aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028889840&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028889840&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Pikovskij, Arkadij 1956- Politi, Antonio 1975- Lyapunov exponents a tool to explore complex dynamics Lyapunov exponents Differential equations Ljapunov-Exponent (DE-588)4123668-3 gnd |
| subject_GND | (DE-588)4123668-3 |
| title | Lyapunov exponents a tool to explore complex dynamics |
| title_auth | Lyapunov exponents a tool to explore complex dynamics |
| title_exact_search | Lyapunov exponents a tool to explore complex dynamics |
| title_full | Lyapunov exponents a tool to explore complex dynamics Arkady Pikovsky, University of Potsdam, Antonio Politi, University of Aberdeen |
| title_fullStr | Lyapunov exponents a tool to explore complex dynamics Arkady Pikovsky, University of Potsdam, Antonio Politi, University of Aberdeen |
| title_full_unstemmed | Lyapunov exponents a tool to explore complex dynamics Arkady Pikovsky, University of Potsdam, Antonio Politi, University of Aberdeen |
| title_short | Lyapunov exponents |
| title_sort | lyapunov exponents a tool to explore complex dynamics |
| title_sub | a tool to explore complex dynamics |
| topic | Lyapunov exponents Differential equations Ljapunov-Exponent (DE-588)4123668-3 gnd |
| topic_facet | Lyapunov exponents Differential equations Ljapunov-Exponent |
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