Crossover of the f(α) Spectra between Critical and Chaotic Regime, and Universal Critical Scaling Laws for Two-Dimensional Fractality : General and Mathematical Physics

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Chaotic attractors of physical systems differ from those of one-dimensional maps due to the high-dimensional fractality. The fractality is described by the spectrum f(α) of singularities of the natural invariant measure. We explore the fractality of the chaotic attractors generated by the period-doubling cascade in the neighborhood of the critical point. Then the attractor consists of a number of thin bands whose number is M=2^m, (m=3, 4, 5, ゥ, and is characterized by the least size l_d of the nearest-neighbor distances between the centers of bands and the least size l_w(<l_d) of the band widths. The distribution of bands gives a spectrum f(α) for the critical regime with length scale l > l_d which approaches the spectrum f_∞(α) at criticality as m→∞. The distribution of chaotic orbits within each band brings about a spectrum f(α) for the chaotic regime with length scale l<l_w which deviates from the chaotic spectrum f_*(α) of the logistic map due to the two-dimensional fractality. As the bifurcation parameter a approaches the critical point a=a_∞, however, the deviation is shown to obey the scaling law ∣a-a_∞∣^κ with κ≡log2/logδ=0.44980ゥalong the band-splitting cascade (a-a_∞)∝δ^<-m>, (m→∞).
理論物理学刊行会の論文
1991-08-25