Characterizing chaos through Lyapunov metrics

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 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 modeled explicitly (analytically) or reconstructed from experimental time-series data. Several practical algorithms for calculating Lyapunov exponents are summarized. Extensions of the Lyapunov exponent approach to studying chaos are also described briefly as they are capable of dealing with the multiscale nature of chaotic signals. The extensions include the Lyapunov fractal dimension, the Kolmogorov--Sinai and Re´nyi entropies, as well as the Re´nyi fractal dimension spectrum and the Mandelbrot fractal singularity spectrum.