- 作 者：（德）坎兹著
- 出 版 社：北京：世界图书北京出版公司
- 出版年份：2015
- ISBN：9787510087721
- 标注页数：369 页
- PDF页数：388 页
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Ⅰ Basic topics 1
1 Introduction:why nonlinear methods? 3
2 Linear tools and general considerations 13
2.1 Stationarity and sampling 13
2.2 Testing for stationarity 15
2.3 Linear correlations and the power spectrum 18
2.3.1 Stationarity and the low-frequency component in the power spectrum 23
2.4 Linear filters 24
2.5 Linear predictions 27
3 Phase space methods 30
3.1 Determinism:uniqueness in phase space 30
3.2 Delay reconstruction 35
3.3 Finding a good embedding 36
3.3.1 False neighbours 37
3.3.2 The time lag 39
3.4 Visual inspection of data 39
3.5 Poincaré surface of section 41
3.6 Recurrence plots 43
4 Determinism and predictability 48
4.1 Sources of predictability 48
4.2 Simple nonlinear prediction algorithm 50
4.3 Verification of successful prediction 53
4.4 Cross-prediction errors:probing stationarity 56
4.5 Simple nonlinear noise reduction 58
5 Instability:Lyapunov exponents 65
5.1 Sensitive dependence on initial conditions 65
5.2 Exponential divergence 66
5.3 Measuring the maximal exponent from data 69
6 Self-similarity:dimensions 75
6.1 Attractor geometry and fractals 75
6.2 Correlation dimension 77
6.3 Correlation sum from a time series 78
6.4 Interpretation and pitfalls 82
6.5 Temporal correlations,non-stationarity,and space time separation plots 87
6.6 Practical considerations 91
6.7 A useful application:determination of the noise level using the correlation integral 92
6.8 Multi-scale or self-similar signals 95
6.8.1 Scaling laws 96
6.8.2 Detrended fluctuation analysis 100
7 Using nonlinear methods when determinism is weak 105
7.1 Testing for nonlinearity with surrogate data 107
7.1.1 The null hypothesis 109
7.1.2 How to make surrogate data sets 110
7.1.3 Which statistics to use 113
7.1.4 What can go wrong 115
7.1.5 What we have learned 117
7.2 Nonlinear statistics for system discrimination 118
7.3 Extracting qualitative information from a time series 121
8 Selected nonlinear phenomena 126
8.1 Robustness and limit cycles 126
8.2 Coexistence of attractors 128
8.3 Transients 128
8.4 Intermittency 129
8.5 Structural stability 133
8.6 Bifurcations 135
8.7 Quasi-periodicity 139
Ⅱ Advanced topics 141
9 Advanced embedding methods 143
9.1 Embedding theorems 143
9.1.1 Whitney's embedding theorem 144
9.1.2 Takens's delay embedding theorem 146
9.2 The time lag 148
9.3 Filtered delay embeddings 152
9.3.1 Derivative coordinates 152
9.3.2 Principal component analysis 154
9.4 Fluctuating time intervals 158
9.5 Multichannel measurements 159
9.5.1 Equivalent variables at different positions 160
9.5.2 Variables with different physical meanings 161
9.5.3 Distributed systems 161
9.6 Embedding of interspike intervals 162
9.7 High dimensional chaos and the limitations of the time delay embedding 165
9.8 Embedding for systems with time delayed feedback 171
10 Chaotic data and noise 174
10.1 Measurement noise and dynamical noise 174
10.2 Effects of noise 175
10.3 Nonlinear noise reduction 178
10.3.1 Noise reduction by gradient descent 179
10.3.2 Local projective noise reduction 180
10.3.3 Implementation of locally projective noise reduction 183
10.3.4 How much noise is taken out? 186
10.3.5 Consistency tests 191
10.4 An application:foetal ECG extraction 193
11 More about invariant quantities 197
11.1 Ergodicity and strange attractors 197
11.2 Lyapunov exponents Ⅱ 199
11.2.1 The spectrum of Lyapunov exponents and invariant manifolds 200
11.2.2 Flows versus maps 202
11.2.3 Tangent space method 203
11.2.4 Spurious exponents 205
11.2.5 Almost two dimensional flows 211
11.3 Dimensions Ⅱ 212
11.3.1 Generalised dimensions,multi-fractals 213
11.3.2 Information dimension from a time series 215
11.4 Entropies 217
11.4.1 Chaos and the flow of information 217
11.4.2 Entropies of a static distribution 218
11.4.3 The Kolmogorov-Sinai entropy 220
11.4.4 The ∈-entropy per unit time 222
11.4.5 Entropies from time series data 226
11.5 How things are related 229
11.5.1 Pesin's identity 229
11.5.2 Kaplan-Yorke conjecture 231
12 Modelling and forecasting 234
12.1 Linear stochastic models and filters 236
12.1.1 Linear filters 237
12.1.2 Nonlinear filters 239
12.2 Deterministic dynamics 240
12.3 Local methods in phase space 241
12.3.1 Almost model free methods 241
12.3.2 Local linear fits 242
12.4 Global nonlinear models 244
12.4.1 Polynomials 244
12.4.2 Radial basis functions 245
12.4.3 Neural networks 246
12.4.4 What to do in practice 248
12.5 Improved cost functions 249
12.5.1 Overfitting and model costs 249
12.5.2 The errors-in-variables problem 251
12.5.3 Modelling versus prediction 253
12.6 Model verification 253
12.7 Nonlinear stochastic processes from data 256
12.7.1 Fokker—Planck equations from data 257
12.7.2 Markov chains in embedding space 259
12.7.3 No embedding theorem for Markov chains 260
12.7.4 Predictions for Markov chain data 261
12.7.5 Modelling Markov chain data 262
12.7.6 Choosing embedding parameters for Markov chains 263
12.7.7 Application:prediction of surface wind velocities 264
12.8 Predicting prediction errors 267
12.8.1 Predictability map 267
12.8.2 Individual error prediction 268
12.9 Multi-step predictions versus iterated one-step predictions 271
13 Non-stationary signals 275
13.1 Detecting non-stationarity 276
13.1.1 Making non-stationary data stationary 279
13.2 Over-embedding 280
13.2.1 Deterministic systems with parameter drift 280
13.2.2 Markov chain with parameter drift 281
13.2.3 Data analysis in over-embedding spaces 283
13.2.4 Application:noise reduction for human voice 286
13.3 Parameter spaces from data 288
14 Coupling and synchronisation of nonlinear systems 292
14.1 Measures for interdependence 292
14.2 Transfer entropy 297
14.3 Synchronisation 299
15 Chaos control 304
15.1 Unstable periodic orbits and their invariant manifolds 306
15.1.1 Locating periodic orbits 306
15.1.2 Stable/unstable manifolds from data 312
15.2 OGY-control and derivates 313
15.3 Variants of OGY-control 316
15.4 Delayed feedback 317
15.5 Tracking 318
15.6 Related aspects 319
A Using the TISEAN programs 321
A.1 Information relevant to most of the routines 322
A.1.1 Efficient neighbour searching 322
A.1.2 Re-occurring command options 325
A.2 Second-order statistics and linear models 326
A.3 Phase space tools 327
A.4 Prediction and modelling 329
A.4.1 Locally constant predictor 329
A.4.2 Locally linear prediction 329
A.4.3 Global nonlinear models 330
A.5 Lyapunov exponents 331
A.6 Dimensions and entropies 331
A.6.1 The correlation sum 331
A.6.2 Information dimension,fixed mass algorithm 332
A.6.3 Entropies 333
A.7 Surrogate data and test statistics 334
A.8 Noise reduction 335
A.9 Finding unstable periodic orbits 336
A.10 Multivariate data 336
B Description of the experimental data sets 338
B.1 Lorenz-like chaos in an NH3 laser 338
B.2 Chaos in a periodically modulated NMR laser 340
B.3 Vibrating string 342
B.4 Taylor-Couette flow 342
B.5 Multichannel physiological data 343
B.6 Heart rate during atrial fibrillation 343
B.7 Human electrocardiogram(ECG) 344
B.8 Phonation data 345
B.9 Postural control data 345
B.10 Autonomous CO2 laser with feedback 345
B.11 Nonlinear electric resonance circuit 346
B.12 Frequency doubling solid state laser 348
B.13 Surface wind velocities 349
References 350
Index 365