We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.

Permutation entropy: a natural complexity measure for time series.

The synchronization of chaotic systems

[1] The analysis of univariate or multivariate time series provides crucial information to describe, understand, and predict climatic variability. The discovery and implementation of a number of novel methods for extracting useful information from time series has recently revitalized this classical field of study. Considerable progress has also been made in interpreting the information so obtained in terms of dynamical systems theory. In this review we describe the connections between time series analysis and nonlinear dynamics, discuss signal-to-noise enhancement, and present some of the novel methods for spectral analysis. The various steps, as well as the advantages and disadvantages of these methods, are illustrated by their application to an important climatic time series, the Southern Oscillation Index. This index captures major features of interannual climate variability and is used extensively in its prediction. Regional and global sea surface temperature data sets are used to illustrate multivariate spectral methods. Open questions and further prospects conclude the review.

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Advanced spectral methods for climatic time series

From the Publisher:
Kalman filtering is a well-established topic in the field of control and signal processing and represents by far the most refined method for the design of neural networks. This book takes a nontraditional nonlinear approach and reflects the fact that most practical applications are nonlinear. The book deals with important applications in such fields as control, financial forecasting, and idle speed control.

Kalman Filtering and Neural Networks

Surrogate time series

Regular Dynamics: Newton to Poincare KAM Theorem | Bifurcations: Routes to Chaos, Stability and Instability | Reconstruction of Phase Space: Regular and Chaotic Motions Observed Chaos | Choosing Time Delays: Chaos as an Information Source Average Mutual Information. | Choosing the Dimension of Reconstructed Phase Space | Invariants of the Motion: Global & Local Lyapunov Exponents Lorenz Model | Modeling Chaos: Local & Global Models Phase Space Models | Signal Separation: Probabilistic Cleaning 'Blind' Signal Separation | Control and Chaos: Parametric Control Examples of Control (including magnetoelastic ribbon, electric circuits, cardiac tissue) | Synchronization of Chaotic Systems: Identical or Dissimilar Systems Chaotic Nonlinear Circuits | Other Example Systems: Laser Intensity Fluctuations Volume Fluctuations of the Great Salt Lake Motion in a Fluid Boundary Layer | Estimating in Chaos: Cramer-Rao Bounds | The Chaos Toolkit: Making 'Physics' out of Chaos

https://www.springer.com/productFlyer_978-0-387-98372-1.pdf?SGWID=0-0-1297-1587867-0

Jerry P. Gollub

Papers

Analysis of Observed Chaotic Data