Dimensions and Entropies in Chaotic Systems : Quantification of Complex Behavior Proceeding of an International Workshop at the Pecos River Ranch, New Mexico, September 11–16, 1985

These proceedings contain the papers contributed to the International Work­ shop on "Dimensions and Entropies in Chaotic Systems" at the Pecos River Conference Center on the Pecos River Ranch in Spetember 1985. The work­ shop was held by the Center for Nonlinear Studies of the Los Alamos National Laboratory. At the Center for Nonlinear Studies the investigation of chaotic dynamics and especially the quantification of complex behavior has a long tradition. In spite of some remarkable successes, there are fundamental, as well as nu­ merical, problems involved in the practical realization of these algorithms. This has led to a series of publications in which modifications and improve­ ments of the original methods have been proposed. At present there exists a growing number of competing dimension algorithms but no comprehensive review explaining how they are related. Further, in actual experimental ap­ plications, rather than a precise algorithm, one finds frequent use of "rules of thumb" together with error estimates which, in many cases, appear to be far too optimistic. Also it seems that questions like "What is the maximal dimension of an attractor that one can measure with a given number of data points and a given experimental resolution?" have still not been answered in a satisfactory manner for general cases. Front Matter....Pages II-IX Front Matter....Pages 1-1 Introductory Remarks....Pages 2-5 Front Matter....Pages 7-7 The Characterization of Fractal Measures as Interwoven Sets of Singularities: Global Universality at the Transition to Chaos....Pages 8-18 Fractal Measures (Their Infinite Moment Sequences and Dimensions) and Multiplicative Chaos: Early Works and Open Problems....Pages 19-27 On the Hausdorff Dimension of Graphs and Random Recursive Objects....Pages 28-33 Chaos-Chaos Phase Transition and Dimension Fluctuation....Pages 34-41 Hausdorff Dimensions for Sets with Broken Scaling Symmetry....Pages 42-53 Scaling in Fat Fractals....Pages 54-60 Front Matter....Pages 61-61 Lorenz Cross-Sections and Dimension of the Double Rotor Attractor....Pages 62-66 On the Fractal Dimension of Filtered Chaotic Signals....Pages 67-73 Efficient Algorithms for Computing Fractal Dimensions....Pages 74-81 Using Mutual Information to Estimate Metric Entropy....Pages 82-91 Front Matter....Pages 93-93 Intermediate Length Scale Effects in Lyapunov Exponent Estimation....Pages 94-99 Comparison of Algorithms for Determining Lyapunov Exponents from Experimental Data....Pages 100-107 A Measure of Chaos for Open Flows....Pages 108-111 Front Matter....Pages 113-113 An Approach to Error-Estimation in the Application of Dimension Algorithms....Pages 114-122 Invisible Errors in Dimension Calculations: Geometric and Systematic Effects....Pages 123-136 Methods for Estimating the Intrinsic Dimsnionality of High-Dimensional Point Sets....Pages 137-147 Front Matter....Pages 149-149 Characterizing Turbulent Channel Flow....Pages 150-157 Characterization of Chaotic Instabilities in an Electron-Hole Plasma in Germanium....Pages 158-170 Instabilities, Turbulence, and the Physics of Fixed Points....Pages 171-178 Front Matter....Pages 179-179 Determination of Attractor Dimension and Entropy for Various Flows: An Experimentalist’s Viewpoint....Pages 180-190 Transition from Quasiperiodicity into Chaos in the Periodically Driven Conductivity of BSN Crystals....Pages 191-197 Dimension and Entropy for Quasiperiodic and Chaotic Convection....Pages 198-206 Experimental Study of the Attractor of a Driven Rayleigh-Bénard System....Pages 207-214 Dimension Measurements from Cloud Radiance....Pages 215-221 Chaos in Open Flow Systems....Pages 222-230 Lasers and Brains: Complex Systems with Low-Dimensional Attractors....Pages 231-240 Evidence of Chaotic Dynamics of Brain Activity During the Sleep Cycle....Pages 241-245 Problems Associated with Dimensional Analysis of Electroencephalogram Data....Pages 246-256 Back Matter....Pages 257-257 This volume contains a collection of papers on methods for the quantification of chaotic dynamical systems. New developments in the theory of nonlinar dynamical systems show that irregular behavior can be generated by deterministic systems with very few degrees of freedom. The concepts of fractal dimensions, dynamical entropies and Lyapunov exponents have been introduced in order to estimate the number of degrees of freedom involved in a given signal or time series. This book provides insight into the mathematical problems of dimensional analysis of erratic data, into the problems of its numerical implementation, and also into its practical realization in a series of different experiments. The limits of predictability of chaotic systems and the reliability and accuracy of different methods for computing dimensions are discussed. New experimental results on spatio-temporal chaos, dimensions of clouds, lasers, brain waves, and hydrodynamical and solid state systems are presented. Introduction General Theory, Mathematical Aspects of Dimensions, Basic Problems Numerical and Experimental Problems in the Calculation of Dimensions and Entropies Computation of Lyapunov Exponents Reliability, Accuracy and Data Requirements of Different Algorithms Analysing Spatio Temporal Chaos Experimental Results and Applications Index of Contributors.