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IMI/Publicaţii/BASM/Ediţii/BASM n.2 (42), 2003/

Algebraic equations with invariant coefficients in qualitative study of the polynomial homogeneous differential systems. (English)

Authors: Baltag Valeriu

Abstract

For planar polynomial homogeneous real vector field X=(P,Q) with deg(P) = deg(Q) = n some algebraic equations of degree n+1 with GL(2,R)-invariant coefficients are constructed. A recurrent method for the construction of these coefficients is given. In the generic case each real or imaginary solution si(i=1,2,..., n+1) of the main equation is a value of the derivative of the slope function, calculated for the corresponding invariant line. Other constructed equations have, respectively, the solutions 1/si, 1-si, si/(si - 1), (si-1)/si, 1/(1-si). The equation with the solutions (n+1)si-1 is called residual equation. If X has real invariant lines, the values and signs of solutions of constructed equations determine the behavior of the orbits in a neighbourhood at infinity. If X has not real invariant lines, it is shown that the necessary and sufficient conditions for the center existence can be expressed through the coefficients of residual equation.

Institute of Mathematics and Computer Science
Academy of Sciences of Moldova,
5 Academiei str. Chisinau,
MD-2028, Republic of Moldova
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