Circuits,signals,and systems
Chaos
Neural networks and natural intelligence
Neural networks and natural intelligence
Practical numerical algorithms for chaotic systems
Practical numerical algorithms for chaotic systems
Foundations of cognitive science
Foundations of cognitive science
Parallel distributed processing: explorations in the microstructure of cognition, vol. 1: foundations
Parallel distributed processing: explorations in the microstructure, vol. 2: psychological and biological models
Chaos: a program collection for the PC
Chaos: a program collection for the PC
Signal processing with fractals: a wavelet-based approach
Signal processing with fractals: a wavelet-based approach
Nonlinear time series analysis
Nonlinear time series analysis
Visualizing chaos: Lyapunov surfaces and volumes
IEEE Computer Graphics and Applications
Chaos and Time-Series Analysis
Chaos and Time-Series Analysis
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Science, engineering, medicine, biology and many other areas deal with signals acquired in the form of time series from different dynamical systems for the purpose of analysis, diagnosis and control of the systems. The signals are often mixed with noise. Separating the noise from the signal may be very difficult if both the signal and the noise are broadband. The problem becomes inherently difficult when the signal is chaotic because its power spectrum is indistinguishable from a broadband noise. This paper describes how to measure and analyze chaos using Lyapunov metrics. The principle of characterizing strange attractors by the divergence and folding of trajectories is studied. A practical approach to evaluating the largest local and global Lyapunov exponents by rescaling and renormalization leads to calculating the m Lyapunov exponents for m-dimensional strange attractors either modelled explicitly (analytically), or reconstructed from experimental time-series data. Several practical algorithms for calculating Lyapunov exponents are summarized. The Lyapunov fractal dimension andKolmogorov-Sinai and Rényi entropies are also described as they are related to the Lyapunov exponents.