Preface to the first editionPreface to the second editionAcknowledgementsI Basic topics1 Introduction: why nonlinear methods? 2 Linear tools and general considerations2.1 Stationarity and sampling2.2 Testing for stationarity2.3 Linear correlations and the power spectrum2.3.1 Stationarity and the low-frequency component in thepower spectrum2.4 Linear filters2.5 Linear predictions3 Phase space methods3.1 Determinism: uniqueness in phase space3.2 Delay reconstruction3.3 Finding a good embedding3.3.1 False neighbours3.3.2 The time lag3.4 Visual inspection of data3.5 Poincar6 surface of section3.6 Recurrence plots 4 Determinism and predictability4.1 Sources of predictability4.2 Simple nonlinear prediction algorithm4.3 Verification of successful prediction4.4 Cross-prediction errors: probing stationarity4.5 Simple nonlinear noise reduction5 Instability: Lyapunov exponents5.1 Sensitive dependence on initial conditions5.2 Exponential divergence5.3 Measuring the maximal exponent from data6 Self-similarity: dimensions6.1 Attractor geometry arid fractals6.2 Correlation dimension6.3 Correlation sum from a time series6.4 Interpretation and pitfalls6.5 Temporal correlations, non-stationarity, and space timeseparation plots6.6 Practical considerations6.7 A useful application: determination of the noise level using thecorrelation integral6.8 Multi-scale or self-similar signals6.8.1 Scaling laws6.8.2 Detrended fluctuation analysis7 Using nonlinear methods when determinism is weak7.1 Testing for nonlinearity with surrogate data7.1.1 The hypothesis7.1.2 How to make surrogate data sets7.1.3 Which statistics to use7.1.4 What can go wrong7.1.5 What we have learned7.2 Nonlinear statistics for system discrimination7.3 Extracting qualitative information from a time series8 Selected nonlinear phenomena8.1 Robustness and limit cycles8.2 Coexistence of attractors8.3 Transients8.4 Intermittency8.5 Structural stability8.6 Bifurcations8.7 Quasi-periodicityII Advanced topics9 Advanced embedding methods9.1 Embedding theorems9.1.1 Whitney's embedding theorem9.1.2 Takens's delay embedding theorem9.2 The time lag9.3 Filtered delay embeddings9.3.1 Derivative coordinates9.3.2 Principal component analysis9.4 Fluctuating time intervals9.5 Multichannel measurements9.5.1 Equivalent variables at different positions9.5.2 Variables with different physical meanings9.5.3 Distributed systems9.6 Embedding of interspike intervals9.7 High dimensional chaos and the limitations of the time delayembedding9.8 Embedding for systems with time delayed feedback10 Chaotic data and noise10.1 Measurement noise and dynamical noise10.2 Effects of noise10.3 Nonlinear noise reduction10.3.1 Noise reduction by gradient descent10.3.2 Local projective noise reduction10.3.3 Implementation of locally projective noise reduction10.3.4 How much noise is taken out? 10.3.5 Consistency tests10.4 An application: foetal ECG extraction11 More about invariant quantities11.1 Ergodicity and strange attractors11.2 Lyapunov exponents II11.2.1 The spectrum of Lyapunov exponents and invariantmanifolds11.2.2 Flows versus maps11.2.3 Tangent space method11.2.4 Spurious exponents11.2.5 Almost two dimensional flows11.3 Dimensions II11.3.1 Generalised dimensions, multi-fractals11.3.2 Information dimension from a time series11.4 Entropies11.4.1 Chaos and the flow of information11.4.2 Entropies of a static distribution11.4.3 The Kolmogorov-Sinai entropy11.4.4 The E-entropy per unit time11.4.5 Entropies from time series data11.5 How things are related11.5.1 Pesin's identity11.5.2 Kaplan-Yorke conjecture12 Modelling and forecasting12.1 Linear stochastic models and filters12.1.1 Linear filters12.1.2 Nonlinear filters12.2 Deterministic dynamics12.3 Local methods in phase space12.3.1 Almost model free methods12.3.2 Local linear fits12.4 Global nonlinear models12.4.1 Polynomials12.4.2 Radial basis functions12.4.3 Neural networks12.4.4 What to do in practice12.5 Improved cost functions12.5.1 Overfitting and model costs12.5.2 The errors-in-variables problem12.5.3 Modelling versus prediction12.6 Model verification12.7 Nonlinear stochastic processes from data12.7.1 Fokker-Planck equations from data12.7.2 Markov chains in embedding space12.7.3 No embedding theorem for Markov chains12.7.4 Predictions for Markov chain data12.7.5 Modelling Markov chain data12.7.6 Choosing embedding parameters for Markov chains12.7.7 Application: prediction of surface wind velocities12.8 Predicting prediction errors12.8.1 Predictability map12.8.2 Individual error prediction12.9 Multi-step predictions versus iterated one-step predictions13 Non-stationary signals13.1 Detecting non-stationarity13.1.1 Making non-stationary data stationary13.2 Over-embedding13.2.1 Deterministic systems with parameter drift13.2.2 Markov chain with parameter drift13.2.3 Data analysis in over-embedding spaces13.2.4 Application: noise reduction for human voice13.3 Parameter spaces from data14 Coupling and synchronisation of nonlinear systems14.1 Measures for interdependence14.2 Transfer entropy14.3 Synchronisation15 Chaos control15.1 Unstable periodic orbits and their invariant manifolds15.1.1 Locating periodic orbits15.1.2 Stable/unstable manifolds from data15.2 OGY-control and derivates15.3 Variants of OGY-control15.4 Delayed feedback15.5 Tracking15.6 Related aspectsA Using the TISEAN programsA.1 Information relevant to most of the routinesA.1.1 Efficient neighbour searchingA.1.2 Re-occurring command optionsA.2 Second-order statistics and linear modelsA.3 Phase space toolsA.4 Prediction and modellingA.4.1 Locally constant predictorA.4.2 Locally linear predictionA.4.3 Global nonlinear modelsA.5 Lyapunov exponentsA.6 Dimensions and entropiesA.6.1 The correlation sumA.6.2 Information dimension, fixed mass algorithmA.6.3 EntropiesA.7 Surrogate data and test statisticsA.8 Noise reductionA.9 Finding unstable periodic orbitsA.10 Multivariate dataB Description of the experimental data setsB.1 Lorenz-like chaos in an NH3 laserB.2 Chaos in a periodically modulated NMR laserB.3 Vibrating stringB.4 Taylor--Couette flowB.5 Multichannel physiological dataB.6 Heart rate during atrial fibrillationB.7 Human electrocardiogram (ECG)B.8 Phonation dataB.9 Postural control dataB.10 Autonomous CO2 laser with feedbackB.11 Nonlinear electric resonance circuitB.12 Frequency doubling solid state laserB.13 Surface wind velocitiesReferencesIndex