
StatusThe thesis was presented on the 25 July, 2008Approved by NCAA on the 18 September, 2008 Abstract– 0.47 Mb / in romanianThesisCZU 517.925
2.25 Mb /
in russian 
In the present thesis we consider a system of differential equations , (1) where are the coefficient tensors, symmetrical in lower indices, where the complete convolution holds, and is a finite set of real nonnegative numbers. In cases when , system (1) has absolute terms. As system (1) is welldefined by the number n and the set , later we will denote it as .
The work is devoted to application of Lie algebras of operators and of the theory of algebraic invariants to differential systems . An infinitedimensional Lie algebra of operators, admitted by system , is considered. It is showed that this algebra has, as a subalgebra, a finitedimensional Lie algebra, corresponding to the representation of centeraffine group GL(n,R) in space of phase variables and coefficients of indicated system. Lie theorem on integrating factor is generalized for ndimensional polynomial differential systems. There are constructed recurrent formulas of Lie operators, corresponding to oneparametrical elementary subgroups of group GL(n,R) in space of phase variables and coefficient of system . The functional basis of mixed comitants is established for system with respect to group GL(4,R). The absence of some dimensions of GL(n,R) orbit is investigated for systems with n=4,5, and Lie algebra of operators, admitted by some canonical forms of indicated systems, is also constructed. An ndimensional Lie algebra of operators is constructed and corresponds to the representation of group GL(n,R) in space of phase variables and coefficients of system with n=4,5, some its characteristics are investigated for n=4. With the aid of Lie algebras of operators, admitted by Darboux type's systems with m= , invariant GL(2,R) integrating factors and invariant particular GL(2,R) integrals were constructed. The invariant particular GL(2,R) integrals in some certain cases are the limit cycles for indicated systems. It is showed that the class of twodimensional polynomial differential systems (m>2), having the equality K2=0 as a centeraffine quadratic integral, is wider then the class of Darboux type's systems, possessing this property. Moreover, obtained centeraffine invariant conditions, ensuring this property, are necessary and sufficient conditions.
Some invariant GL(n,R) integrals for affine systems with n=4,5 are considered. For Darboux type's system (n>2) there are constructed recurrent formulas for some invariant GL(n,R) integrals. System is considered and its invariant GL(3,R) integrals are found. For the same system, there are constructed examples with particular integrals, having
the form of ellipsoid or sphere. Systems with n=4,5, with n=3,4 and are investigated in order to construct their invariant particular integrals.