Correlations in one-dimensional fully developed chaos |
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Authors: | Takashi Tsuchiya Atsushi Ichimura Yoshinori Nagai |
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Affiliation: | (1) Biophysics Division, Natural Science Laboratory, Advance Co. Ltd., 2-8-18 Ohashi, Meguro-ku, 153 Tokyo, Japan;(2) Department of Applied Physics, Waseda University, 160 Tokyo, Japan;(3) Laboratory of Molecular Biology, School of Veterinary Medicine, Azabu University, Fuchinobe, 229 Sagamihara, Japan;(4) Present address: Department of Management Information, Yamanashi Gakuni University, 2-4-5 Sakaori, 400 Kofu, Japan |
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Abstract: | Correlations in the baker map and the tent map as examples of one-dimensional, fully developed chaos are considered. It is shown, utilizing symbolic dynamical systems derived from these maps, that the vanishing second-order correlation function is not sufficient to guarantee uncorrelatedness. Importance of the higher-order, especially third-order, correlation functions is emphasized for chaotic systems. In search of the quantities that grasp correlational behaviors as a whole in chaotic systems, it is proposed to use the fixed-separation correlation integral, which is a modified quantity of the usual correlation integral devised to calculate the fractal dimension of strange attractors, for these maps. It is shown that the new quantity contains all the even-number orders of autocorrelation function that are commonly considered. |
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Keywords: | One-dimensional chaos higher-order correlation function correlation integral |
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