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Physics | Analysis of Observed Chaotic Data

Analysis of Observed Chaotic Data

Abarbanel, Henry

1st ed. 1996. 2nd printing 1997, XIV, 272 p.

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  • About this textbook

When I encountered the idea of chaotic behavior in deterministic dynami­ cal systems, it gave me both great pause and great relief. The origin of the great relief was work I had done earlier on renormalization group properties of homogeneous, isotropic fluid turbulence. At the time I worked on that, it was customary to ascribe the apparently stochastic nature of turbulent flows to some kind of stochastic driving of the fluid at large scales. It was simply not imagined that with purely deterministic driving the fluid could be turbulent from its own chaotic motion. I recall a colleague remarking that there was something fundamentally unsettling about requiring a fluid to be driven stochastically to have even the semblance of complex motion in the velocity and pressure fields. I certainly agreed with him, but neither of us were able to provide any other reasonable suggestion for the observed, apparently stochastic motions of the turbulent fluid. So it was with relief that chaos in nonlinear systems, namely, complex evolution, indistinguish­ able from stochastic motions using standard tools such as Fourier analysis, appeared in my bag of physics notions. It enabled me to have a physi­ cally reasonable conceptual framework in which to expect deterministic, yet stochastic looking, motions. The great pause came from not knowing what to make of chaos in non­ linear systems.

Content Level » Graduate

Related subjects » Physics

Table of contents 

Regular Dynamics: Newton to Poincaré; 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: Cramér-Rao Bounds | The Chaos Toolkit: Making 'Physics' out of Chaos

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