Statistical description of chaotic attractors: The dimension function

Abstract

A method for the investigation of fractal attractors is developed, based on statistical properties of the distribution P(δ, n) of nearest-neighbor distances δ between points on the attractor. A continuous infinity of dimensions, called dimension function, is defined through the moments of P(δ, n). In particular, for the case of self-similar sets, we prove that the dimension function DF yields, in suitable points, capacity, information dimension, and all other Renyi dimensions. An algorithm to compute DF is derived and applied to several attractors. As a consequence, an estimate of nonuniformity in dynamical systems can be performed, either by direct calculation of the uniformity factor, or by comparison among various dimensions. Finally, an analytical study of the distribution P(δ, n) is carried out in some simple, meaningful examples. © 1985 Plenum Publishing Corporation.