FRACTAL RADIOPHYSICS. 1. THEORETICAL BASES

Abstract

Purpose: Currently, there is a tendency to “fractalize” the science. Radiophysics is no exception. The subject of this work is a review of the basic ideas of “fractalization”, the mathematical foundations of modern fractal methods for describing and exploring the world. The purpose of the work is to present the basic concepts, definitions and relationships of the modern theory of fractals, as well as the classification and analysis of existing numerical characteristics of fractals. Design/methodology/approch: The methods of constructing geometric monofractals and multifractals are considered. A comparative characteristic of the methods for assessing the dimension of physical fractals is given. Examples of physical fractals are given. Findings: In the development of the “fractalization” of science, 4 stages are distinguished: the era of “monsters”, the preparatory stage, the stage of formation and development, the modern stage. For the correct description of fractals, the Hausdorff–Besicovitch dimension, which can also take noninteger values, is used. The following fractal classifications are considered: mathematical and physical, geometric and algebraic, mono- and multifractals, regular and stochastic, homogeneous and heterogeneous. It has been demonstrated that the fractal dimension of objects can be both fractional and integer, it is important that the fractal dimension should be greater than their topological dimension. The equality of the fractal dimensions of two objects does not imply the similarity of their structure. When describing thick fractals as regular monofractals, instead of the Hausdorff–Besicovitch dimension, the scaling exponents are used. Conclusions: The mathematical foundations of the theory of fractals, used in the modern theoretical radiophysics, are presented.

Keywords

• fractal
• fractal dimension
• fractal classification
• monofractal
• multifractal
• multifractal formalism
• types of dimension
• scaling