Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8408Research ArticleSynthesis of 3D-dynamical Systems with Critical Points of Given Topological StructuresVolkovS VDepartment of nonlinear analysis and optimizationsvlvolkov@rambler.ruPeoples’ Friendship University of Russia150320133112008092016Copyright © 2013,2013The problem of synthesis of normal autonomous systems of ordinary differential equations which three-dimensional phase spaces have isolated equilibrium points with desired topolog- ical structure properties. To solve this problem a method based on the using special vector fields of comparison directions is proposed. While choosing these vector fields it should be taken into account that the local structure of an isolated equilibrium point is completely characterized by: a) a set of singular phase trajectories and surfaces that break up the neigh- borhood of the equilibrium point into elementary areas, and b) behavior of non-singular phase trajectories in these areas. Thus obtained vector fields allow, under certain conditions, to present the local topological structure properties of equilibrium point in an analytical form as algebraic expressions with respect to phase coordinates. These expressions are used to set up the equations equal in number to the number of dimensions of the phase space and which are the algebraic equations with respect to the right-hand sides of sought differential equations. The main purpose of the paper is to describe the general approach to the posed problem, so the solution is considered only in one particular case where all the elementary areas of the sought dynamical system equilibrium point are elementary areas of one of the possible types. Theoretical results of the article are illustrated by a concrete example. Presented in this paper is a partial generalization of the previously published results for solving inverse problems of the theory of dynamical systems on the plane.dynamical systemsystem of differential equationsphase spacesequilibrium pointsphase space topological structurescritical pointsseparatrix sur- facesvector fields of comparison directionsдинамические системысистемы дифференциальных уравненийфазовые пространствасостояния равновесиятопологические структуры разбиения на траекторииособые поверхностивекторные поля направлений сравнения