Fractal-cluster theory and thermodynamic principles of the control and analysis for the self-organizing systems

Abstract

The theory of resource distribution in self-organizing systems on the basis of the fractal-cluster method has been presented. In turn, the fractal-cluster method is based on the fractal-cluster relations of V.P. Burdakov and the analytical apparatus of the thermodynamics of I. Prigozhin's structure. This theory consists of two parts: deterministic and probabilistic. The first part includes the static and dynamic criteria, the fractal-cluster dynamic equations which are based on the Fibonacci's range characteristics fractal-cluster correlations. The second part includes fundamentals of the probabilistic theory of a fractal-cluster systems. This part includes the dynamic equations of the probabilistic evolution of these systems. By using the numerical researches of these equations for the stationary case the random state field of the self-organizing system in the phase space of the $D$, $H$ criteria and probability $f$ have been obtained. For the socio-economical and biological systems this theory has been tested. In particular, three fundamental fractal-cluster laws have been obtained for biological organisms: probabilistic, energetic and evolutionary.