NUMERICAL APPROXIMATION OF THE PERCENTAGE OF ORDER FOR ONE-DIMENSIONAL MAPS

Abstract

The percentage of organized motion of the chaotic zone (which shall from now on be referred to as percentage of order) for the logistic, the sine-square and the 4-exponent map, is calculated. The calculations are reached via a sampling method that incorporates the Lyapunov exponent. Although these maps are specially selected examples of one-dimensional ones, the conclusions can also be applied to any other one-dimensional map. Since the metric characteristics of a bifurcation diagram of a unimodal map, such as the referred percentage of order, are dependent on the order of the maximum, this dependence is verified for several maps. Once the chaotic zone can be separated into regions between the sequential band mergings, the percentage of order corresponding to each region is calculated for the logistic map. In each region, the resultant area occupied by order, or the supplementary area occupied by chaos, participates in a sequence similar to Feigenbaum's one, which converges to the same respective Feigenbaum's constant.