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Author: Natalia Gherştega Degree: doctor of physics and mathematics Speciality:  01.01.02 - Differential equations Year: 2007 Scientific adviser: Mihail Popa doctor habilitat, professor, Institute of Mathematics and Computer Science Institution: Scientific council:

Status The thesis was presented on the 23 February, 2007 Approved by NCAA on the 19 April, 2007 Abstract – 0.29 Mb / in romanian – 0.38 Mb / in russian Thesis CZU 517.925 0.94 Mb / in russian 133 pages

Keywords

Summary

In the thesis is considered the system of type

dxj/dt=Xk2Aajj1j2...jkxj1xj2...xjk (j, j1, j2, ..., jk = 1, 3), (1)

where coefficients tensor a^ • • (fc G A) - are symmetrical in lower indexes in which the complete convolution takes place, A is a finite set of the different positive integers.

The work is devoted to application of Lie algebras of operators, mixed comitants and the theory of orbits in investigation of the system of type (1). The defining equations for n-dimensional polynomial differential systems are built. With the aid of their it is shown that the Lie algebras of operators L9, connected with centro-affine group GL(3, E) admits by the system (1) and the criterion of invariance for invariants and mixed comitants for the three-dimensional differential system with respect to this group is hold. Lie theorem on integrating factor for the three-dimensional polynomial is hold and the relation with the equations of Pfaff it is shown. For the system of the type (1) with A = {0, 1}, {0,2}, {1,2}, {0, 1,2} the functional bases of centro-affine invariants are obtaned.The bases of centro-affine comitants, contravariants, covariants of the lower order for the system (1) with A = {I}, {0,1}, {2}, {1,2} are investigated. In some cases the necessary and sufficient invariant conditions are formulates that dim^,O(a) = 9 in classification of GL(3,R) — orbit of the affine system (1) with A = {0, 1}. The classification of GL(3,R) — orbit for linear system (1) with A = {1} is constructed completely. Lie algebras of operators admits of the canonical type of linear, affine differential system and the differential system of Darboux type are obtained. With the aid of these algebras and Lie theorem on integrating factor in the three-dimensional case the first integrals of this system on the GL(3,R) — orbit of maximal dimensions and some invariant expressions of GL(3,R) — integrals are constructed. In general case the invariant particular GL(3, E) — integrals of these systems on GL(3,E)— orbits of maximal dimensions 9 are obtained. It is considerated the presence for the three-dimensional differential system of Darboux type with quadratic nonlinearities linear particular integrals. The algebraic invariant particular GL(3,E)— integrals for three-dimensional differential system of Darboux type with cubic nonlinearities are investigated.

The thesis is written in Russian language.

Oficial Reviewers

• Vladimir Dragan
  doctor, associate professor (docent), Technical University of Moldova
• Anton Sadovskii
  doctor habilitat în ştiinţe fizico-matematice,

 Theses