With the advent of electronic computers, numerical simulations of dynamical models have
become an increasingly appreciated way to study complex and nonlinear systems. This has
been accompanied by an evolution of theoretical tools and concepts: some of them, more
suitable for a pure mathematical analysis, happened to be less practical for applications;
other techniques proved instead very powerful in numerical studies, and their popularity
exploded. Lyapunov exponents is a perfect example of a tool that has flourished in the
modern computer era, despite having been introduced at the end of the nineteenth century.
The rigorous proof of the existence of well-defined Lyapunov exponents requires subtle
assumptions that are often impossible to verify in realistic contexts (analogously to other
properties, e.g., ergodicity). On the other hand, the numerical evaluation of the Lyapunov
exponents happens to be a relatively simple task; therefore they are widely used in many
setups. Moreover, on the basis of the Lyapunov exponent analysis, one can develop novel
approaches to explore concepts such as hyperbolicity that previously appeared to be of
purely mathematical nature.
In this book we attempt to give a panoramic view of the world of Lyapunov exponents,
from their very definition and numerical methods to the details of applications to various
complex systems and phenomena. We adopt a pragmatic, physical point of view, avoiding
the fine mathematical details. Readers interested in more formal mathematical aspects are
encouraged to consult publications such as the recent books by Barreira and Pesin (2007)
and Viana (2014).
Author(s): Arkady Pikovsky, Antonio Politi
Publisher: Cambridge University Press
Year: 2016
Language: English
Pages: 298
Contents......Page 6
Preface......Page 11
1.1.1 Early results......Page 14
1.1.2 Biography of Aleksandr Lyapunov......Page 16
1.1.3 Lyapunov’s contribution......Page 17
1.1.4 The recent past......Page 18
1.2 Outline of the book......Page 19
1.3 Notations......Page 21
2.1 The mathematical setup......Page 23
2.2 One-dimensional maps......Page 24
2.3 Oseledets theorem......Page 25
2.3.1 Remarks......Page 26
2.3.2 Oseledets splitting......Page 28
2.3.3 “Typical perturbations” and time inversion......Page 29
2.4.1 Stability of fixed points and periodic orbits......Page 30
2.5.1 Deterministic vs. stochastic systems......Page 31
2.5.2 Relationship with instabilities and chaos......Page 32
2.5.3 Invariance......Page 33
2.5.4 Volume contraction......Page 34
2.5.5 Time parametrisation......Page 35
2.5.6 Symmetries and zero Lyapunov exponents......Page 37
2.5.7 Symplectic systems......Page 39
3.1 The largest Lyapunov exponent......Page 41
3.2 Full spectrum: QR decomposition......Page 42
3.2.2 Householder reflections......Page 44
3.3 Continuous methods......Page 46
3.4 Ensemble averages......Page 48
3.5 Numerical errors......Page 49
3.5.1 Orthogonalisation......Page 50
3.5.2 Statistical error......Page 51
3.5.3 Near degeneracies......Page 53
3.6 Systems with discontinuities......Page 56
3.6.1 Pulse-coupled oscillators......Page 61
3.6.2 Colliding pendula......Page 62
3.7 Lyapunov exponents from time series......Page 63
4 Lyapunov vectors......Page 67
4.1 Forward and backward Oseledets vectors......Page 68
4.2 Covariant Lyapunov vectors and the dynamical algorithm......Page 70
4.3 Dynamical algorithm: numerical implementation......Page 72
4.4 Static algorithms......Page 74
4.4.2 Kuptsov-Parlitz algorithm......Page 75
4.5 Vector orientation......Page 76
4.6 Numerical examples......Page 77
4.7 Further vectors......Page 78
4.7.1 Bred vectors......Page 79
4.7.2 Dual Lyapunov vectors......Page 80
5.1 Finite-time analysis......Page 83
5.2 Generalised exponents......Page 86
5.3 Gaussian approximation......Page 90
5.4.1 Quick tools......Page 91
5.4.2 Weighted dynamics......Page 92
5.5 Eigenvalues of evolution operators......Page 93
5.6 Lyapunov exponents in terms of periodic orbits......Page 97
5.7.1 Deviation from hyperbolicity......Page 102
5.7.2 Weak chaos......Page 103
5.7.3 Hénon map......Page 107
5.7.4 Mixed dynamics......Page 109
6.1 Lyapunov exponents and fractal dimensions......Page 113
6.2 Lyapunov exponents and escape rate......Page 116
6.3 Dynamical entropies......Page 118
6.4.1 Generalised Kaplan-Yorke formula......Page 120
6.4.2 Generalised Pesin formula......Page 122
7.1 Finite vs. infinitesimal perturbations......Page 123
7.2 Computational issues......Page 125
7.2.1 One-dimensional maps......Page 127
7.3 Applications......Page 128
8 Random systems......Page 131
8.1.1 Weak disorder......Page 132
8.1.2 Highly symmetric matrices......Page 138
8.1.3 Sparse matrices......Page 141
8.1.4 Polytomic noise......Page 144
8.2.1 First-order stochastic model......Page 149
8.2.2 Noise-driven oscillator......Page 150
8.2.3 Khasminskii theory......Page 154
8.2.4 High-dimensional systems......Page 155
8.3.1 LEs as eigenvalues and supersymmetry......Page 159
8.3.2 Weak-noise limit......Page 162
8.3.3 Synchronisation by common noise and random attractors......Page 163
9.1 Coupling sensitivity......Page 165
9.1.1 Statistical theory and qualitative arguments......Page 166
9.1.2 Avoided crossing of LEs and spacing statistics......Page 170
9.1.3 A statistical-mechanics example......Page 172
9.1.4 The zero exponent......Page 173
9.2.1 Complete synchronisation and transverse Lyapunov exponents......Page 175
9.2.2 Clusters, the evaporation and the conditional Lyapunov exponent......Page 176
9.2.3 Synchronisation on networks and master stability function......Page 177
10.1 Lyapunov density spectrum......Page 181
10.1.1 Infinite systems......Page 184
10.2 Chronotopic approach and entropy potential......Page 186
10.3 Convective exponents and propagation phenomena......Page 191
10.3.1 Mean-field approach......Page 194
10.3.2 Relationship between convective exponents and chronotopic analysis......Page 196
10.3.3 Damage spreading......Page 198
10.4.1 Hamiltonian systems......Page 200
10.4.2 Differential-delay models......Page 204
10.4.3 Long-range coupling......Page 206
11.1 Lyapunov dynamics as a roughening process......Page 213
11.1.1 Relationship with the KPZ equation......Page 215
11.1.2 The bulk of the spectrum......Page 220
11.2 Localisation of the Lyapunov vectors and coupling sensitivity......Page 222
11.3 Macroscopic dynamics......Page 226
11.3.1 From micro to macro......Page 229
11.3.2 Hydrodynamic Lyapunov modes......Page 231
11.4 Fluctuations of the Lyapunov exponents in space-time chaos......Page 232
11.5 Open system approach......Page 236
11.5.2 Scaling behaviour of the invariant measure......Page 239
12.1 Anderson localisation......Page 242
12.2 Billiards......Page 244
12.3.1 Escape rate......Page 248
12.3.2 Molecular dynamics......Page 249
12.4 Lagrangian coherent structures......Page 250
12.5 Celestial mechanics......Page 252
12.6 Quantum chaos......Page 255
A.1 Lumped systems: discrete time......Page 258
A.2 Lumped systems: continuous time......Page 259
A.3 Lattice systems: discrete time......Page 260
A.4 Lattice systems: continuous time......Page 261
A.5 Spatially continuous systems......Page 262
A.8 Global coupling: continuous time......Page 263
Appendix B Pseudocodes......Page 265
C.1 Gaussian matrices: discrete time......Page 269
C.2 Gaussian matrices: continuous time......Page 270
Appendix D Symbolic encoding......Page 271
Bibliography......Page 272
Index......Page 290
