Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On the Solutions, Integrals and Limit Cycles of N Darboux's System

Author(s):

V. N. Gorbuzov

Grodno State University
230023, Grodno, Ozheshko st., 22
Belarus

gorbuzov@grsu.unibel.by

P. B. Pauliuchyk

Grodno State University
230023, Grodno, Ozheshko st., 22
Belarus

gorbuzov@grsu.unibel.by

Abstract:

For differential Darboux's system dw/dz=[a(z)+M(z,w)E]w^T, where a(z)=|a_ij(z)|, n ≥ 2, E is identity matrix, a_ij a complex holomorphic functions, M(w,z) homogeneous function of degree ρ of w with coefficients which are holomorphic functions of z the questions of the integrability and the absence of movable critical singular points of its solutions are considered. In case of autunomous Darboux's system it is fulfiled the building of the first integrals and last jacobi's multiplier on the base of which general integral is constructed. Also the theorems of quantity and the algebraicity of limit cycles are proved.

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