Natural processes existing in complex objects of inanimate and living matter are of a stochastic and non-equilibrium nature. The main problem in the study of such systems is to determine the value of entropy as a quantitative measure of the uncertainty and systematicity of states of dynamical systems in different phase spaces. This paper presents a new method for analyzing active processes of solar dynamics using the theory of non-Markov random discrete processes (NMRDP). The NMRDP theory is based on the Zwanzig-Mori kinetic equations in a finite-difference discrete interpretation. This is consistent with the concept of non-equilibrium statistical condensed matter physics. Qualitative information about the set of behavioral patterns, relaxation processes, dynamic characteristics and internal properties of solar activity can be obtained using NMRDP modeling by the author’s methodological approach developed in this work. This approach is focused on the analysis of spectral frequency memory functions, dynamic orthogonal parameters, phase transformations, relaxation and kinetic processes and self-organization in complex physical systems. In this work, for modeling NMRDP, the author’s software package APSASA (automated program for solar activity stochastic analysis) was used, which also allows predicting the trend of solar activity for a limited period of time. Modeling NMRDP associated with active processes occurring on the Sun made it possible to build a mathematical model with whose help it is possible to study the regularities and randomness of stochastic processes, as well as to reveal the patterns arising from the recurrence and periodicity of solar activity.

In this work, the projective geometry method was used for analyzing star clusters. When carrying out the calculation procedures, it was considered that non-linear distortion factors had been removed from the measured stars’ coordinates. Determination of stars’ proper motions is of great practical importance, as the inertial coordinate system relies on catalogues of star positions, and it is necessary to be aware of the stellar reference marks’ time shift. In the practical part of the work, the breadboard simulation of the use of the proposed method for determining stars’ proper motions is performed. At the same time, it is supposed that at 90”/mm breadboard image scale the absolute values of proper motions do not exceed 0.050” over a period of 50 years. As result, determined that the standard deviation of the calculated proper motions μα , μβ from their true value is 0.0065 arcseconds for the first model (when the proper motions of the reference “stars” are negligible and equal to 0) and 0.0072 arcseconds or the second model (when the reference stars do have real proper motions). These values indicate the high accuracy of the used method.