Preface ....................................................... vii
1. The phenomenology of complex systems ......................... 1
1.1. Complexity, a new paradigm .............................. 1
1.2. Signatures of complexity ................................ 3
1.3. Onset of complexity ..................................... 5
1.4. Four case studies ....................................... 8
1.4.1. Rayleigh-Benard convection ....................... 8
1.4.2. Atmospheric and climatic variability ............ 11
1.4.3. Collective problem solving: food recruitment
in ants ......................................... 15
1.4.4. Human systems ................................... 19
1.5. Summing up ............................................. 23
2. Deterministic view .......................................... 25
2.1. Dynamical systems, phase space, stability .............. 25
2.1.1. Conservative systems ............................ 27
2.1.2. Dissipative systems ............................. 27
2.2. Levels of description .................................. 34
2.2.1. The microscopic level ........................... 34
2.2.2. The macroscopic level ........................... 36
2.2.3. Thermodynamic formulation ....................... 38
2.3. Bifurcations, normal forms, emergence .................. 41
2.4. Universality, structural stability ..................... 46
2.5. Deterministic chaos .................................... 49
2.6. Aspects of coupling-induced complexity ................. 53
2.7. Modeling complexity beyond physical science ............ 59
3. The probabilistic dimension of complex systems .............. 64
3.1. Need for a probabilistic approach ...................... 64
3.2. Probability distributions and their evolution laws ..... 65
3.3. The retrieval of universality .......................... 72
3.4. The transition to complexity in probability space ...... 77
3.5. The limits of validity of the macroscopic
description ............................................ 82
3.5.1. Closing the moment equations in the
mesoscopic description .......................... 82
3.5.2. Transitions between states ...................... 84
3.5.3. Average values versus fluctuations in
deterministic chaos ............................. 88
3.6. Simulating complex systems ............................. 90
3.6.1. Monte Carlo simulation .......................... 91
3.6.2. Microscopic simulations ......................... 92
3.6.3. Cellular automata ............................... 94
3.6.4. Agents, players and games ....................... 95
3.7. Disorder-generated complexity .......................... 96
4. Information, entropy and selection ......................... 101
4.1. Complexity and information ............................ 101
4.2. The information entropy of a history .................. 104
4.3. Scaling rules and selection ........................... 106
4.4. Time-dependent properties of information.
Information entropy and thermodynamic entropy ......... 115
4.5. Dynamical and statistical properties of time
histories. Large deviations, fluctuation theorems ..... 117
4.6. Further information measures. Dimensions and
Lyapunov exponents revisited .......................... 120
4.7. Physical complexity, algorithmic complexity,
and computation ....................................... 124
4.8. Summing up: towards a thermodynamics of complex
systems ............................................... 128
5. Communicating with a complex system: monitoring,
analysis and prediction .................................... 131
5.1. Nature of the problem ................................. 131
5.2. Classical approaches and their limitations ............ 131
5.2.1. Exploratory data analysis ...................... 132
5.2.2. Time series analysis and statistical
forecasting .................................... 135
5.2.3. Sampling in time and in space .................. 138
5.3. Nonlinear data analysis ............................... 139
5.3.1. Dynamical reconstruction ....................... 139
5.3.2. Symbolic dynamics from time series ............. 143
5.3.3. Nonlinear prediction ........................... 148
5.4. The monitoring of complex fields ...................... 151
5.4.1. Optimizing an observational network ............ 153
5.4.2. Data assimilation ............................... 157
5.5. The predictability horizon and the limits of
modeling .............................................. 159
5.5.1. The dynamics of growth of initial errors ....... 160
5.5.2. The dynamics of model errors ................... 164
5.5.3. Can prediction errors be controlled? ........... 170
5.6. Recurrence as a predictor ............................. 171
5.6.1. Formulation .................................... 172
5.6.2. Recurrence time statistics and dynamical
complexity ..................................... 176
5.7. Extreme events ........................................ 180
5.7.1. Formulation .................................... 180
5.7.2. Statistical theory of extremes ................. 182
5.7.3. Signatures of a deterministic dynamics in
extreme events ................................. 185
5.7.4. Statistical and dynamical aspects of the
Hurst phenomenon ............................... 191
6. Selected topics ............................................ 195
6.1. The arrow of time ..................................... 195
6.1.1. The Maxwell-Boltzmann revolution, kinetic
theory, Boltzmann's equation ................... 196
6.1.2. First resolution of the paradoxes: Markov
processes, master equation ..................... 200
6.1.3. Generalized kinetic theories ................... 202
6.1.4. Microscopic chaos and nonequilibrium
statistical mechanics .......................... 204
6.2. Thriving on fluctuations: the challenge of being
small ................................................. 208
6.2.1. Fluctuation dynamics in nonequilibrium steady
states revisited ............................... 210
6.2.2. The peculiar energetics of irreversible
paths joining equilibrium states ............... 211
6.2.3. Transport in a fluctuating environment far
from equilibrium ............................... 214
6.3. Atmospheric dynamics .................................. 217
6.3.1. Low order models ............................... 218
6.3.2. More detailed models ........................... 222
6.3.3. Data analysis .................................. 223
6.3.4. Modeling and predicting with probabilities ..... 224
6.4. Climate dynamics ...................................... 226
6.4.1. Low order climate models ....................... 227
6.4.2. Predictability of meteorological versus
climatic fields ................................ 230
6.4.3. Climatic change ................................ 233
6.5. Networks .............................................. 235
6.5.1. Geometric and statistical properties of
networks ....................................... 236
6.5.2. Dynamical origin of networks ................... 239
6.5.3. Dynamics on networks ........................... 244
6.6. Perspectives on biological complexity ................. 247
6.6.1. Nonlinear dynamics and self-organization at
the biochemical, cellular and organismic
level .......................................... 249
6.6.2. Biological superstructures ..................... 251
6.6.3. Biological networks ............................ 253
6.6.4. Complexity and the genome organization ......... 260
6.6.5. Molecular evolution ............................ 263
6.7. Equilibrium versus nonequilibrium in complexity and
self-organization ..................................... 267
6.7.1. Nucleation ..................................... 268
6.7.2. Stabilization of nanoscale patterns ............ 272
6.7.3. Supramolecular chemistry ....................... 274
6.8. Epistemological insights from complex systems ......... 276
6.8.1. Complexity, causality and chance ............... 277
6.8.2. Complexity and historicity ..................... 279
6.8.3. Complexity and reductionism .................... 283
6.8.4. Facts, analogies and metaphors ................. 285
Color plates .................................................. 287
Suggestions for further reading ............................... 291
Index ......................................................... 321
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