Research On M-J Fractal Sets Based On Equipotential Point Algorithm

Mandelbrot sets (M sets) and Julia sets (J sets) are the most typical sets among fractal theory which have been applied in our practical life after being established and developed. A great number of researches have been done on fractal theory. It has been applied in many interdisciplinary subjects, thus also improves the development of fractal theory.Features of M-J sets, the typical fractal sets, as well as structural characteristics and evolution laws of them under random noise perturbation are and will always be the hot topics today and in the future. On the basis of the pre-existing algorithm, this paper put forward the Equipotential Point Algorithm (EPA), by which a J sets is iterated in the complex plane under f:z←z2+c. This paper analyzes deeply on iteration performance of points in J sets, studies on the features of M-J sets under noise perturbation and also discusses the correlation between them.This paper reviews the history and current present of fractal theory, and then introduces its basic theories and algorithms. EPA is an improvement of Escape Time Algorithm. According to EPA, J sets is drawn on the basis of struck frequency of the points in the screen, and different struck numbers are given different colors. In this paper, J sets are created by using EPA in a complex plane and its features are analyzed by dividing the set into interior points and exterior points. To further study on it, the iteration process of individual point in the screen was also analyzed. Experimental results show that in the internal region of J sets, points in main petals are struck more frequently and are hierarchically distributed; in the external region, points closer to the crevice of main petals are struck more seriously; points in other parts are seldom struck or escaped more rapidly.On the basis of EPA, M sets and J sets under addictive noise perturbation and multiplicative noise perturbation are created respectively in this paper. We also analyze the changes of M sets and J sets under random noise perturbation and discuss the possible relationship between them. Experimental results indicate that M sets and J sets iterated under the same mapping Zn+1=Zn2+c are inextricably linked; even if given the same perturbation parameter, M sets and J sets show differences in stability.