Synthesis of 3D-dynamical Systems with Critical Points of Given Topological Structures

The 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 system, system of differential equations, phase spaces, equilibrium points, phase space topological structures, critical points, separatrix sur- faces, vector fields of comparison directions