CROSSOVER OF THE F(A) SPECTRA BETWEEN CRITICAL AND CHAOTIC REGIME, AND UNIVERSAL CRITICAL SCALING LAWS FOR 2-DIMENSIONAL FRACTALITY

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(alpha) 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 = 2m, (m = 3, 4, 5, ...), 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(alpha) for the critical regime with length scale 1 > l(d) which approaches the spectrum f infinity(alpha) at criticality as m --> infinity. The distribution of chaotic orbits within each band brings about a spectrum f(alpha) for the chaotic regime with length scale l < l(w) which deviates from the chaotic spectrum f*(alpha) of the logistic map due to the two-dimensional fractality. As the bifurcation parameter a approaches the critical point a = a infinity, however, the deviation is shown to obey the scaling law \a-a infinity\kappa with kappa = log2/log-delta = 0.44980... along the band-splitting cascade (a-a infinity) is-proportional-to delta-m, (m --> infinity).