Call number:
AWI S2-98-0216
In:
Cambridge nonlinear science series, 7
Description / Table of Contents:
Deterministic chaos offers a striking explanation for irregular behaviour and anomalies in systems which do not seem to be inherently stochastic. The most direct link between chaos theory and the real world is the analysis of time series from real systems in terms of nonlinear dynamics. This book provides experimentalists with methods for processing, enhancing, and analysing measured signals using these methods; and for theorists it also demonstrates the practical applicability of mathematical results. The framework of deterministic chaos constitutes a new approach to the analysis of irregular time series. Traditionally, nonperiodic signals have been modelled by linear stochastic processes. But even very simple chaotic dynamical systems can exhibit strongly irregular time evolution without random inputs. Chaos theory offers completely new concepts and algorithms for time series analysis which can lead to a thorough understanding of the signals. The book introduces a broad choice of such concepts and methods, including phase space embeddings, nonlinear prediction and noise reduction, Lyapunov exponents, dimensions and entropies, as well as statistical tests for nonlinearity. Related topics such as chaos control, wavelet analysis, and pattern dynamics are also discussed. Applications range from high-quality, strictly deterministic laboratory data to short, noisy sequences which typically occur in medicine, biology, geophysics, and the social sciences. All the material discussed is illustrated using real experimental data. This book will be of value to any graduate student or researcher who needs to be able to analyse time series data, especially in the fields of physics, chemistry, biology, geophysics, medicine, economics, and the social sciences.
Type of Medium:
Monograph available for loan
Pages:
XVI, 304 Seiten
,
Illustrationen
,
25 cm
Edition:
First published
ISBN:
0-521-55144-7
Series Statement:
Cambridge nonlinear science series 7
URL:
https://www.gbv.de/dms/goettingen/219381496.pdf
Language:
English
Note:
Preface
Acknowledgements
Part S Basic topics
Chapter I Introduction: Why nonlinear methods?
Chapter 2 Linear tools and general considerations
2.1 Stationarity and sampling
2.2 Testing for stationarity
2.3 Linear correlations and the power spectrum
2.3.1 Stationarity and the low-frequency component in the power spectrum
2.4 Linear filters
2.5 Linear predictions
Chapter 3 Phase space methods
3.1 Determinism: Uniqueness in phase space
3.2 Delay reconstruction
3.3 Finding a good embedding
3.4 Visual inspection of data
3.5 Poincare surface of section
Chapter 4 Determinism and predictability
4.1 Sources of predictability
4.2 Simple nonlinear prediction algorithm
4.3 Verification of successful prediction
4.4 Probing stationarity with nonlinear predictions
4.5 Simple nonlinear noise reduction
Chapter 5 Instability: Lyapunov exponents
5.1 Sensitive dependence on initial conditions
5.2 Exponential divergence
5.3 Measuring the maximal exponent from data
Chapter 6 Self-similarity: Dimensions
6.1 Attractor geometry and fractals
6.2 Correlation dimension
6.3 Correlation sum from a time series
6.4 Interpretation and pitfalls
6.5 Temporal correlations, nonstationarity, and space time separation plots
6.6 Practical considerations
6.7 A useful application: Determination of the noise level
Chapter 7 Using nonlinear methods when determinism is weak
7.1 Testing for nonlinearity with surrogate data
7.1.1 The null hypothesis
7.1.2 How to make surrogate data sets
7.1.3 Which statistics to use
7.1.4 What can go wrong
7.1.5 What we have learned
7.2 Nonlinear statistics for system discrimination
7.3 Extracting qualitative information from a time series
Chapters Selected nonlinear phenomena
8.1 Coexistence of attractors
8.2 Transients
8.3 Intermittency
8.4 Structural stability
8.5 Bifurcations
8.6 Quasi-periodicity
Part 2 Advanced topics
Chapter 9 Advanced embedding methods
9.1 Embedding theorems
9.1.1 Whitney's embedding theorem
9.1.2 Takens's delay embedding theorem
9.2 The time lag
9.3 Filtered delay embeddings
9.3.1 Derivative coordinates
9.3.2 Principal component analysis
9.4 Fluctuating time intervals
9.5 Multichannel measurements
9.5.1 Equivalent variables at different positions
9.5.2 Variables with different physical meanings
9.5.3 Distributed systems
9.6 Embedding of interspike intervals
Chapter 10 Chaotic data and noise
10.1 Measurement noise and dynamical noise
10.2 Effects of noise
10.3 Nonlinear noise reduction
10.3.1 Noise reduction by gradient descent
10.3.2 Local projective noise reduction
10.3.3 Implementation of locally projective noise reduction
10.3.4 How much noise is taken out?
10.3.5 Consistency tests
10.4 An application: Foetal ECG extraction
Chapter ! 1 More about invariant quantities
11.1 Ergodicity and strange attractors
11.2 Lyapunov exponents II
11.2.1 The spectrum of Lyapunov exponents and invariant manifolds
11.2.2 Flows versus maps
11.2.3 Tangent space method
11.2.4 Spurious exponents
11.2.5 Almost two-dimensional flows
11.3 Dimensions II
11.3.1 Generalised dimensions, multifractals
11.3.2 Information dimension from a time series
11.4 Entropies
11.4.1 Chaos and the flow of information
11.4.2 Entropies of a static distribution
11.4.3 The Kolmogorov-Sinai entropy
11.4.4 Entropies from time series data
11.5 How things are related
11.5.1 Pesin's identity
11.5.2 Kaplan-Yorke conjecture
Chapter 12 Modelling and forecasting
12.1 Stochastic models
12.1.1 Linear filter
12.1.2 Nonlinear filters
12.1.3 Markov models
12.2 Deterministic dynamics
12.3 Local methods in phase space
12.3.1 Almost model free methods
12.3.2 Local linear fits
12.4 Global nonlinear models
12.4.1 Polynomials
12.4.2 Radial basis functions
12.4.3 Afeura/ networks
12.4.4 Wfcat to do in practice
12.5 Improved cost functions
12.5.1 Overfitting and model costs
12.5.2 The errors-in-variables problem
12.6 Model verification
Chapter 13 Chaos control
13.1 Unstable periodic orbits and their invariant manifolds
13.1.1 Locating periodic orbits
13.1.2 Stable/unstable manifolds from data
13.2 OGY-control and derivates
13.3 Variants of OGY-control
13.4 Delayed feedback
13.5 Chaos suppression without feedback
13.6 Tracking
13.7 Related aspects
Chapter 14 Other selected topics
14.1 High dimensional chaos
14.1.1 Analysis of higher dimensional signals
14.1.2 Spatially extended systems
14.2 Analysis of spatiotemporal patterns
14.3 Multiscale or self-similar signals, wavelets
14.3.1 Dynamical origin of multiscale signals
14.3.2 Scaling laws
14.3.3 Wavelet analysis
Appendix A Efficient neighbour searching
Appendix B Program listings
Appendix C Description of the experimental data sets
References
Index
Location:
AWI Reading room
Branch Library:
AWI Library
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