Yu.I. Nechaev - Dr. Sc. (Eng.), Professor, State University of Information Technology, Mechanic and Optic, St.-Petersburg, Russia, Honored Scientist of RF. E-Mail: nechaev@ifmo.mail.ru

The formal instrument of research of neuro-dynamic systems (ND-systems) topology on the basis of methods of the modern catastrophe theory is discussed. An interpretation of topological structures of ND-systems on the basis of the bifurcation theory and methods of the analysis of nonlinear dissipation systems is of principle concern. A ND-systems representation within the framework of fractal geometry is considered. An interpretation of ND-systems behavior is realized using a model of a choice and a competition principle in the conditions of uncertainty and incompleteness of the initial information. Topological synthesis of ND-systems in complex dynamic environments is carried out with the help of cognitive modeling and the entropy analysis. The formal model of topological dynamics of ND-system is realized on the basis of the program complex functioning in the multiprocessor computing environment on the basis of the operating module. The conceptual model of transformation of the information at interpretation of topological dynamics of ND-systems on the basis of the modern catastrophe theory includes set of strategy of management; set of the elements realizing a principle of a competition. The set defining model of development of menacing situations and set of elements of an operative database (a vector of entrance influences, a condition of an environment and set of laws in data) defines rules of generalization of the information. Analytical and geometrical interpretation of topological dynamics of ND-systems is realized using competing computing technologies. The analysis of alternatives and search of the preferable decision are realized by means of choice model. The topological structure of decisions interpretation is realized on the basis of axiomatics of the sets theory and a paradigm of information processing in the multiprocessor computing environment. Methodological principles of dynamic catastrophe model are substantially expressed by the block of statements and definitions. ND-system evolution is displayed on a basis of synergetic paradigms. Within the limits of such interpretation the approach realized using the generalized principle of dissipation formulated by N.N.Moiseev. Decomposition of state space forms a certain orientation of self-organizing processes and formation of target attractors depending on features of dynamic catastrophe model. This property of nonlinear dissipation systems indicates the self-organizing effect that opens new possibilities of the risks analysis of decision-making at interpretation of extreme situations. Formation of operating influences leads to change of ND-system behavior according to management strategy. The analysis is conducted in a real time mode on the basis of conceptual decisions. The research problem is represented by fractal structure in the form of an ellipse. Formation areas of fractals structures in state space are defined as the limited configuration of set of all conditions of dynamic system in the course of its evolution in the non-stationary environment. Basic difference of the offered approach to information processing consists in use of the integrated models of the analysis and interpretation of dynamic situations. Realization of models of ND-systems is defined by complex character of the mathematical description of infringements of a continuity of investigated phenomena. Advantages of the offered approach consist in the possibility of realization of geometrical and analytical interpretation of ND-systems dynamics, and also in representation of the information in the form of simple geometrical images. Interpretation of topological structures is carried out on the basis of the modern approach to studying of ND-systems at continuous change of dynamics of object in the conditions of uncertainty and incompleteness of the initial information. The developed method of researches has demanded updating of initial theoretical basis of topology and the catastrophe theory: theorems, axioms, definitions and topology statements, with reference to challenges of interaction of ND-systems are formulated. Thus fundamental results of topology are kept in that part, how much it demands the developed mathematical apparatus of ND-systems in the course of evolution in the complex dynamic environment.

 

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