Dimensions and Entropies in Chaotic Systems

1. Front Matter

   Pages II-IX

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2. Introduction

   1. Front Matter

      Pages 1-1

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   2. Introductory Remarks

      • G. Mayer-Kress

      Pages 2-5

3. General Theory, Mathematical Aspects of Dimensions, Basic Problems

   1. Front Matter

      Pages 7-7

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   2. The Characterization of Fractal Measures as Interwoven Sets of Singularities: Global Universality at the Transition to Chaos

      • I. Procaccia

      Pages 8-18

   3. Fractal Measures (Their Infinite Moment Sequences and Dimensions) and Multiplicative Chaos: Early Works and Open Problems

      • B. B. Mandelbrot

      Pages 19-27

   4. On the Hausdorff Dimension of Graphs and Random Recursive Objects

      • R. D. Mauldin

      Pages 28-33

   5. Chaos-Chaos Phase Transition and Dimension Fluctuation

      • Y. Aizawa

      Pages 34-41

   6. Hausdorff Dimensions for Sets with Broken Scaling Symmetry

      • D. K. Umberger, G. Mayer-Kress, E. Jen

      Pages 42-53

   7. Scaling in Fat Fractals

      • J. D. Farmer

      Pages 54-60

4. Numerical and Experimental Problems in the Calculation of Dimensions and Entropies

   1. Front Matter

      Pages 61-61

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   2. Lorenz Cross-Sections and Dimension of the Double Rotor Attractor

      • E. J. Kostelich, J. A. Yorke

      Pages 62-66

   3. On the Fractal Dimension of Filtered Chaotic Signals

      • R. Badii, A. Politi

      Pages 67-73

   4. Efficient Algorithms for Computing Fractal Dimensions

      • F. Hunt, F. Sullivan

      Pages 74-81

   5. Using Mutual Information to Estimate Metric Entropy

      • A. M. Fraser

      Pages 82-91

5. Computation of Lyapunov Exponents

   1. Front Matter

      Pages 93-93

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   2. Intermediate Length Scale Effects in Lyapunov Exponent Estimation

      • A. Wolf, J. A. Vastano

      Pages 94-99

   3. Comparison of Algorithms for Determining Lyapunov Exponents from Experimental Data

      • J. A. Vastano, E. J. Kostelich

      Pages 100-107

   4. A Measure of Chaos for Open Flows

      • R. J. Deissler, K. Kaneko

      Pages 108-111

6. Reliability, Accuracy and Date-Requirements of Different Algorithms

   1. Front Matter

      Pages 113-113

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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.

• algorithms
• chaos
• complex system
• complex systems
• dynamical systems
• entropy
• system
• turbulence

• Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, USA

  Gottfried Mayer-Kress

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