I. Naumeyko, M. Alja’afreh

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Abstract. The dynamic models of the complex ergatic objects’ behavior, presented in the form of differential equations and their systems were studied. The stability and other properties are researched. The methods of analysis and reduce of harmful factors and their impact on people were theoretically proved. The methods of analysis and critical points removal in dynamic models of hazards distribution are offered. The object of study is the system of the harmful external factors protection. Subject of research is the system of two nonlinear differential equations as a model of technical systems with protection. The object of protection is described by logistic equation. and defense system – by non-linear differential equation with a security functions of rather general form. This paper describes critical modes analysis and stationary states’ stability of protected systems with harmful influences. Numerical solution of general problem and also the analytical solution for the case of fixed expected harmful effects have been obtained. Various types of general models for “Man-machine-environment” systems were studied. Each of describes some kind of the practically important quality of object in an appropriate way. And all together they describe the object in terms of it’s safe operation. Their further detailing process results to either well-known, or some new subsystems’ models. Systems with “fast” protection at a relatively slow dynamics of the object were studied. This leads to the models with small parameter and asymptotic solutions of differential equations. Some estimates for protection cost in different price-functional and for different functions in the right part of equation, which describes the dynamics of defense were obtained. For calculations, analysis and graphical representations some of mathematical packages was applied.

Key words: Non-linear system, singular points, eigenvalues, asymptotic behavior, first approximation, linearization.


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