THE ALGORITHM OF THE NONLINEAR DYNAMICS MODELS STUDY

Mamedova Tat'yana Fanadovna, Candidate of physical and mathematical sciences, professor, sub-department of applied mathematics, Mordovia State University named after N. P. Ogarev
(68 Bol'shevistskaya street, Saransk, Russia), mamedovatf@yandex.ru
Lyapina Anna Aleksandrovna, Postgraduate student, Mordovia State University named after N. P. Ogarev
(68 Bol'shevistskaya street, Saransk, Russia), lyapina@e-mordovia.ru

517.9

Background. Nowadays, a lot of real processes are described by nonlinear differential equations. In this regard, one of the most important problems in mathematical modeling is the problem of studying stability of equilibrium states and asymptotic properties of such systems solutions. There is necessity of mathematical apparatus associated with nonlinear systems of differential equations to describe various processes in dynamical systems. Therefore, it is necessary to develop meth-ods of studying such systems and to create new effective methods of the analysis. We are faced with the challenge of analysing nonlinear systems that should allow us to determine the conditions of sustainable operation. It is also essential to develop mathematical methods to study these systems. The purpose of this paper is to con-struct an algorithm of studying the Volterra type equations by using the comparison method; to study the Volterra type mathematical models which are based on nonlinear systems of ordinary differential equations; to conduct numerical experiments based on the algorithm of studying the Volterra type equations.
Materials and methods. Mathematical models of nonlinear dynamics are investigated by using the E.V.Voskresensky’s comparison method, which is a generalization of the Ye. Lyapunov’s functions method and is an effective method of studying dynamic processes. In this paper, we investigate nonlinear differential equations in the following way. The comparison equation is built for the original equation. It is assumed that we know how to solve the comparison equation. Next, using the method of standard functions we compare the two solutions of the equations. The proper selection of the equation of comparison and the reference comparison function allows us to solve various problems of the qualitative theory of differential equations, to study the behavior of differential equations solutions and, what is more important, allows us to solve problems in the theory of stability in critical cases.
Results. The algorithm of studying systems of the Volterra type equations by using the comparison method has been developed. A qualitative study of mathematical models of the Volterra type equations by the comparison method has been done.
Conclusions. The computational algorithm based on the E. V. Voskresensky’s comparison method has been developed. The algorithm has been implemented for the Volterra type model. The asymptotic stability of the equations system in regard to variables has been stated. The outcomes of the study of the "predator-prey" model stability are similar to the findings presented by the authors Yuejian Jie, Yuan Yuan.

the system of ordinary differential equations, asymptotic stability in re-gard to variables, the "predator-prey" model.

1. Voskresenskiy E. V. Asimptoticheskie metody: Teoriya i prilozheniya [Asymptotic methods: Theory and application]. Saransk: Srednevolzhskoe matematicheskoe ob-shchestvo, 2001, 300 p.
2. Mamedova T. F., Desyaev E. V., Lyapina A. A. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physics and mathematics sciences]. 2012, no. 2, pp. 98–105.
3. Yuejian Jie. Yuan Yuan. International Journal of Biology. 2009, vol. 1, no. 2, pp. 22–25.