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.

Klappentext

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.

Inhalt
I Introduction.- Introductory Remarks.- II General Theory, Mathematical Aspects of Dimensions, Basic Problems.- The Characterization of Fractal Measures as Interwoven Sets of Singularities: Global Universality at the Transition to Chaos.- Fractal Measures (Their Infinite Moment Sequences and Dimensions) and Multiplicative Chaos: Early Works and Open Problems.- On the Hausdorff Dimension of Graphs and Random Recursive Objects.- Chaos-Chaos Phase Transition and Dimension Fluctuation.- Hausdorff Dimensions for Sets with Broken Scaling Symmetry.- Scaling in Fat Fractals.- III Numerical and Experimental Problems in the Calculation of Dimensions and Entropies.- Lorenz Cross-Sections and Dimension of the Double Rotor Attractor.- On the Fractal Dimension of Filtered Chaotic Signals.- Efficient Algorithms for Computing Fractal Dimensions.- Using Mutual Information to Estimate Metric Entropy.- IV Computation of Lyapunov Exponents.- Intermediate Length Scale Effects in Lyapunov Exponent Estimation.- Comparison of Algorithms for Determining Lyapunov Exponents from Experimental Data.- A Measure of Chaos for Open Flows.- V Reliability, Accuracy and Data-Requirements of Different Algorithms.- An Approach to Error-Estimation in the Application of Dimension Algorithms.- Invisible Errors in Dimension Calculations: Geometric and Systematic Effects.- Methods for Estimating the Intrinsic Dimensionality of High-Dimensional Point Sets.- VI Analysing Spatio Temporal Chaos.- Characterizing Turbulent Channel Flow.- Characterization of Chaotic Instabilities in an Electron-Hole Plasma in Germanium.- Instabilities, Turbulence, and the Physics of Fixed Points.- VII Experimental Results and Applications.- Determination of Attractor Dimension and Entropy for Various Flows: An Experimentalist'sViewpoint.- Transition from Quasiperiodicity into Chaos in the Periodically Driven Conductivity of BSN Crystals.- Dimension and Entropy for Quasiperiodic and Chaotic Convection.- Experimental Study of the Attractor of a Driven Rayleigh-Bénard System.- Dimension Measurements from Cloud Radiance.- Chaos in Open Flow Systems.- Lasers and Brains: Complex Systems with Low-Dimensional Attractors.- Evidence of Chaotic Dynamics of Brain Activity During the Sleep Cycle.- Problems Associated with Dimensional Analysis of Electroencephalogram Data.- Index of Contributors.