By: Henry DI Abarbanel
272 pages, Figs, 2 tabs
This book develops a clear and systematic treatment of time series of data, regular and chaotic, that one finds in observations of nonlinear systems. The reader is led from measurements of one or more variables through the steps of building models of the source as a dynamical system, classifying the source by its dynamical characteristics, and finally predicting and controlling the dynamical system. The text examines methods for separating the signal of physical interest from contamination by unwanted noise, and for investigating the phase space of the chaotic signal and its properties. The emphasis throughout is on the use of the modern mathematical tools for investigating chaotic behavior to uncover properties of physical systems. The methods require knowledge of dynamical systems at the advanced undergraduate level and some knowledge of Fourier transforms and other signal processing methods. The toolkit developed in the book will provide the reader with efficient and effective methods for analyzing signals from nonlinear sources; these methods are applicable to problems of control, communication, and prediction in a wide variety of systems encountered in physics, chemistry, biology, and geophysics.
Regular Dynamics: Newton to Poincare; KAM Theorem o Bifurcations:Routes to Chaos, Stability and Instability o Reconstruction of PhaseSpace: Regular and Chaotic Motions; Observed Chaos o Choosing TimeDelays: Chaos as an Information Source; Average Mutual Information. oChoosing the Dimension of Reconstructed Phase Space o Invariants ofthe Motion: Global & Local Lyapunov Exponents; Lorenz Model o ModelingChaos: Local & Global Models; Phase Space Models o Signal Separation:Probabilistic Cleaning; "Blind" Signal Separation o Control and Chaos:Parametric Control; Examples of Control (including magnetoelasticribbon, electric circuits, cardiac tissue) o Synchronization ofChaotic Systems: Identical or Dissimilar Systems; Chaotic NonlinearCircuits o Other Example Systems: Laser Intensity Fluctuations; VolumeFluctuations of the Great Salt Lake; Motion in a Fluid Boundary Layero Estimating in Chaos: Cramer-Rao Bounds o The Chaos Toolkit: Making"Physics" out of Chaos
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