Estimating multifractal measures of strange attractors

This paper presents a study of approximations of multifractal measures of strange attractors through the Renyi dimension. The study is based on the probability of each volume element (vel) intersected by the points on the strange attractor. Since the complete strange attractor consists of an infinite number of points, we cannot obtain the theoretical value of the probability; instead, we consider a finite number of points in the vels. Therefore, this study reduces to a finite number of points and finite size of vels. We have shown that, for a given vel size, the Renyi dimension is sensitive to the number of points used in the attractor, and that for a given number of points in the strange attractor, it is also sensitive to the vel size. We also find that for a given vel size, there is a minimum bound on the number of points required. The smaller the vel size, the larger the minimum bound. Furthermore, when the number of points which is above the minimum bound increases, we can see the convergence property of the Renyi dimension. The convergence can be a guideline to determine the number of points required to compute the dimension