book

Deterministic chaos provides a novel framework for the analysis of irregular time series. Traditionally, nonperiodic signals are modeled 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 signal. 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. Also related topics like chaos control, wavelet analysis and pattern dynamics are discussed. Applications range from high quality, strictly deterministic laboratory data to short, noisy sequences which typically occur in medicine, biology, geophysics or the social sciences. All material is discussed and illustrated using real experimental data. For the main algorithms, sample computer programs in C and FORTRAN are given. For the convenience of our readers, the sample programs can also be downloaded from this server.

• Introduction: Why nonlinear methods?
• Linear tools and general considerations
  • Stationarity and sampling
  • Testing for stationarity
  • Linear correlations and the power spectrum
  • Linear filters
  • Linear predictions
• Phase space methods
  • Determinism: Uniqueness in phase space
  • Delay reconstruction
  • Finding a good embedding
  • Visual inspection of data
  • Poincaré surface of section
• Determinism and predictability
  • Sources of predictability
  • Simple nonlinear prediction algorithm
  • Verification of successful prediction
  • Probing stationarity with nonlinear predictions
  • Simple nonlinear noise reduction
• Instability: Lyapunov exponents
  • Sensitive dependence on initial conditions
  • Exponential divergence
  • Measuring the maximal exponent from data
• Self-similarity: Dimensions
  • Attractor geometry and fractals
  • Correlation dimension
  • Correlation sum from a time series
  • Interpretation and pitfalls
  • Temporal correlations, space time separation plot
  • Practical considerations
  • Determination of the noise level
• Using nonlinear methods when determinism is weak
  • Testing for nonlinearity with surrogate data
  • Nonlinear statistics for system discrimination
  • Extracting qualitative information from a time series
• Selected nonlinear phenomena
  • Coexistence of attractors
  • Transients
  • Intermittency
  • Structural stability
  • Bifurcations
  • Quasi-periodicity

• Advanced embedding methods
  • Embedding theorems
  • The time lag 
  • Filtered delay embeddings
  • Fluctuating time intervals
  • Multichannel measurements 
  • Embedding of interspike intervals 
• Chaotic data and noise
  • Measurement noise and dynamical noise
  • Effects of noise
  • Nonlinear noise reduction
  • An application: foetal ECG extraction
• More about invariant quantities
  • Ergodicity and strange attractors
  • Lyapunov exponents II
  • Dimensions II
  • Entropies
  • How things are related
• Modelling and forecasting 
  • Stochastic models
  • Deterministic dynamics
  • Local methods in phase space
  • Global nonlinear models
  • Improved cost functions
  • Model verification 
• Chaos control
  • Unstable periodic orbits and their invariant manifolds
  • OGY-control and derivates
  • Variants of OGY-control
  • Delayed feedback
  • Chaos suppression without feedback
  • Tracking
  • Related aspects
• Other selected topics
  • High dimensional chaos
  • Analysis of spatiotemporal patterns
  • Multiscale or self--similar signals, wavelets

   

• Efficient neighbour searching
• Program listings (C and FORTRAN)
• Description of the experimental data sets