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3.5.2. Popular class of model equations – oscillators................................................................ 106
3.5.3. “Standard from” of ordinary differential equations ........................................................ 109
3.6. Models – discrete maps .............................................................................................. 111
3.6.1. Introduction..................................................................................................................... 111
3.6.2. Exemplary nonlinear maps.............................................................................................. 111
3.6.3. Role of discrete models................................................................................................... 116
3.7. Models of spatially extended systems ........................................................................ 122
3.7.1. Coupled map lattices ....................................................................................................... 123
3.7.2. Cellular automata ............................................................................................................ 128
3.7.3. Delay differential equations ............................................................................................ 130
3.7.4. Partial differential equations ...........................................................................................131
3.8. Artificial neural networks........................................................................................... 132
3.8.1. Standard formal neuron...................................................................................................133
3.8.2. Architecture and classification of neural networks ......................................................... 135
3.8.3. Basic properties and solvable tasks................................................................................. 136
3.8.4. Learning .......................................................................................................................... 137
Chapter 4. Stochastic models of evolution ......................................................................... 139
4.1. Elements of theory of random processes.................................................................... 139
4.1.1. Concept of random process............................................................................................. 139
4.1.2. Characteristics of random process .................................................................................. 140
4.1.3. Stationarity and ergodicity of random processes ............................................................ 141
4.1.4. Statistical estimates of characteristics of random processes ........................................... 143
4.2. Base models of random processes .............................................................................. 143
4.3. Evolutionary equations for probability distribution laws........................................... 146
4.4. Autoregression – moving average processes.............................................................. 147
4.5. Stochastic differential equations and white noise ...................................................... 151
PART II. MODELING FROM TIME SERIES
Chapter 5. Problem settings in modeling from data series .............................................. 155
5.1. Scheme for process of model construction from time series...................................... 155
5.2. Systematization of problems according to amount of a priori information............... 157
5.3. Specific features of empirical modeling problems ..................................................... 158
5.3.1. Direct and inverse problems............................................................................................ 158
5.3.2. Well-posed and ill-posed problems................................................................................. 159
5.3.3. Ill-conditioned problems .................................................................................................161
Chapter 6. Observed series – source of data for modeling............................................... 162
6.1. Observable and model quantities................................................................................ 162
6.1.1. Observations and measurements..................................................................................... 162
6.1.2. Methods to increase and decrease number of characterizing quantities ......................... 166
6.2. Analogous-to-digital converters ................................................................................. 168
6.3. Time series.................................................................................................................. 170
6.3.1. Terms .............................................................................................................................. 170
6.3.2. Examples......................................................................................................................... 171
6.4. Elements of time series analysis................................................................................. 176
6.4.1. Visual express-analysis ................................................................................................... 176
6.4.2. Methods of spectral analysis (Fourier analysis and wavelets) ........................................ 178
6.4.3. Phase of signal and empirical mode decomposition ....................................................... 189
6.4.4. Analysis of stationarity/nonstationarity .......................................................................... 194
6.4.5. Analysis of cross-dependence (correlation, coherence, synchronization) ...................... 195
6.5. Experimental example instead of conclusions ........................................................... 197
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