Tkachuk A. Qualitative connection between properties of so-lutions of differential and corresponding difference equations

The dissertation is devoted to studying difference equations and to identifying conditions of qualitative correspondence between solutions of differential and corresponding difference equations. The limitation of differential and corresponding difference equation solu-tion is investigated. There are established the conditions under which the existance of periodical solution of dif-ferential equation is kept in place, if the solution of corre-sponding difference equation had the following property. The connections between dissipative of differential and difference systems is established, namely the conditions are described which from the dissipative of correspond-ing differential equation follows the dissipative of the difference equation. For the difference equations system the conditions of existance and stability of invariant set is pointed. The relation between invariant sets of differen-tial and difference equations systems is investigated, namely the conditions are described which show the existance of invariant set of differential equations system in terms of constant sighs Lyapunov functions of differ-ence equations corresponding system. The existance of positive definite Lyapunov function is investigated as well, but its derivative is not positive definite for the differential equations system which has smooth, stable and invariant set and set of zeroes of this function coin-cides with the data manifold.