Purpose. The purpose of the article is to justify the theoretical basis and methodological foundations of the systems' transformation in the context of the influence of external and internal factors.
Methodology. The results of the article were obtained through the application of such methods: analysis, synthesis, deduction, induction in the study of systems, the theory of self-organization and scientific concepts. These methods form the basis of a modern study of the self-organization of systems of living or inanimate nature of artificial or natural organic or inorganic, socio-economic, humanitarian, military, political and other origins. Systematic method – for substantiating the laws and principles of the transformation of systems, the general law of transformation of systems; modeling – for developed mathematical model of the system.
Results. The author substantiates the regularities and principles of system' transformation, determines the external and internal factors of this process, reveals their objectivity and subjectivity; the general law of transformation of systems is defined and formulated; An approach to mathematical modeling of systems is proposed on the basis of complex consideration and application of external and internal factors.
Originality. The general law of systems transformation is definite for the first time. The law objectively takes into account all existing theories and concepts concerning transformation, self-organization and modernization of systems, that is quantitative, qualitative and functional changes taking place with them. The mathematical model of the system is represented by the function of the action and interaction of objective and subjective external and internal factors.
Practical value. The obtained research results allow at the present stage of development of science to create a universal approach and the corresponding equipment for modeling, analysis and evaluation of the degree of transformation of systems of any origin. The determination of their productivity, stability, prevention of chaos, dissipation and corresponding corrective actions, depending on missions and goals, actions that are being persecuted and carried out or are taking place. According to the research, the way to the solution of this problem lies in the field of the most optimal representation of the action of external and internal factors by physical, social or any other nature by mathematical, functional, statistical, expert methods on the basis of the corresponding presented functions and qualimetric approaches.