INFLUENCE OF THE STATIONARY AND PERTURBED STATE OF THE CENTRAL FORCE FIELD ON THE FRACTAL CHARACTERISTICS OF THE ATTRACTOR
Authors: Ivan Grabar, Olga Grabar
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The paper states that the known algorithms for generating and constructing fractal sets can be significantly expanded through the family of new algorithms proposed by the authors. These algorithms are based on modelling the attractors of motion of a material point in the field N of central forces in a discrete formulation. When only one of these forces is accidentally switched on at any given time, the point attractor has a strictly fractal structure. It is shown that the perturbation of one or more of the N central forces leads to a change in the structure of the attractor. Thus, the areas of the attractor Dp , controlled by the perturbed forces, with an increase in the perturbation radius, evolve to the perturbation trajectory. For biharmonic perturbations, it is shown that these subsets belong to the inner region of the 2n–point. It has been established that for small values of the perturbation radius R the parameter n → ∞, and for large values of R the parameter n → 1. For the field of central forces in the form of matrices 2*2; 3*3; 5*5 the quantitative models n(2R/B; m) are constructed and their close correlation with the perturbation parameter R, the size of the side B of the square matrix of the field of central forces and the “gravitational” parameter m is shown. It is shown that the gnoseology of the proposed algorithms originates from the wellknown algorithm of M. Barnsley, but the physical and software components are significantly improved and developed. The proposed family of algorithms allows to expand the possibilities of generating original (exclusive) fractal sets up to ~ 1040… 1050 pieces. At the same time, it is possible to control the fractal dimension, porosity, specific gravity, aerodynamic and hydraulic resistance, noise, sound and thermal insulation properties, colour of individual subregions, etc. in a wide range of values. It is shown that a significant part of such fractal sets, especially those with a high degree of symmetry, can be useful for solving problems in the field of design, ergonomics and aesthetics, for decorating buildings, clothing, footwear, haberdashery, toys, as well as for creating puzzles, IQ-tests, etc.