By G. Mayer-Kress (auth.), Dr. Gottfried Mayer-Kress (eds.)

These lawsuits include the papers contributed to the foreign paintings­ store on "Dimensions and Entropies in Chaotic structures" on the Pecos River convention heart at the Pecos River Ranch in Spetember 1985. The paintings­ store was once held by means of the heart for Nonlinear experiences of the Los Alamos nationwide Laboratory. on the middle for Nonlinear experiences the research of chaotic dynamics and particularly the quantification of complicated habit has an extended culture. regardless of a few notable successes, there are primary, in addition to nu­ merical, difficulties desirous about the sensible awareness of those algorithms. This has resulted in a chain of courses during which differences and enhance­ ments of the unique tools were proposed. at the moment there exists an increasing number of competing measurement algorithms yet no accomplished assessment explaining how they're comparable. extra, in genuine experimental ap­ plications, instead of an exact set of rules, one unearths common use of "rules of thumb" including blunders estimates which, in lots of situations, seem to be a long way too confident. additionally it appears questions like "What is the maximal measurement of an attractor that it is easy to degree with a given variety of info issues and a given experimental resolution?" have nonetheless now not been spoke back in a passable demeanour for basic cases.

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Indeed, in many cases the structure of the chaotic attractor is not uniform. 9 Dimension fluctuation (TO = Tp) 40 60 A When the chaos-chaos transition really occurs, one can expect that the enhancement of fluctuations is observed because of the strong non-uniformity of the attractor. In the case of the generalized baker's transformation, the structure of the attractor" is always uniform, so that the fluctuation level is zero. This implies that the observed singularity is quite deceptive and no chaos-chaos transition.

Is defined by 40 The fluctuation of the local exponent The fluctuation is the function of the observation time TO, and when TO goes to infinity the fluctuation becomes zero. However, the dynamical or transient fluctuation can be detected during the finite time observation. The same idea is used for the local dimension parameter DA(TO,XO) defined by, and its fluctuation, Though the fluctuation defined above is the parameter that describes the reliability in measuring each statistical quantity, the important point is that these fluctuations characterize some structural non-uniformity of an attractor.

Transactions American Mathematical SOCiety. (to appear) . 9. Moran. " Proceedings of the Cambridge Phil. SOCiety 42 (1946). 15-23. San 33 Chaos-Chaos Phase Transition and Dimension Fluctuation Y. Aizawa Department of Physics, University of Kyoto, Japan 1. I ntroduct ion Recent studies on chaos have made clear that the concept of chaos is quite different from the probabilistic randomness. Much work has been especially done to understand the internal order in chaos such as topological and fractal ones.

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