GENERAL FORM OF SUPERUNIVERSALITY FOR FRACTAL DIMENSIONS IN ONE-DIMENSIONAL MAPS
Abstract
In one-dimensional continuous maps, there is an internal relationship between equivalent scaling factors αe and fractal dimensions d of Feigenbaum-type attractors on all critical (accumulation) points of transitions to chaos. In general, for m-modal maps with m turning points C1,C2,…,Cm, by defining αe as the geometric average of all scaling factors αCi(i=1,2,…,m), |αe| =|αC1 αC2… αCm|1/m, the superuniversal relationship d(W)log|W| |αe(W)|=β(e) holds, where |W| is the basic period of the m-tuply superstable sequences W, and β(e) is not only independent of the concrete sequences W, but also independent of the concrete maps. This is a general superuniversality for arbitrary multimodal maps, which is verified numerically for the cases of m ≤ 3 and can also be extended to Lorenz maps with a discontinuous point.
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