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IMCS/Publications/BASM/Issues/BASM n.2(90), 2019/

The problem of the center for cubic differential systems with the line at infinity and an affine real invariant straight line of total algebraic multiplicity five

Authors: Şubă Alexandru, Silvia Turuta

Abstract

In this article, we study the real planar cubic differential systems with a non-degenerate monodromic critical point M 0 . In the cases when the algebraic multiplicity m(Z) = 5 or m(l1) + m(Z) ≥ 5, where Z = 0 is the line at infinity and l1 = 0 is an affine real invariant straight line, we prove that the critical point M0 is of the center type if and only if the first Lyapunov quantity vanishes. More over, if m(Z) = 5 (respectively, m(l1) + m(Z) ≥ 5, m(l1) ≥ j, j = 2,3) then M0 is a center if the cubic systems have a polynomial first integral (respectively, an integrating factor of the form 1/l1j).

Vladimir Andrunachievici Institute of Mathematics
and Computer Science
5 Academiei str., Chisinau, MD 2028, Moldova
Tiraspol State University
5 Gh. Iablocichin str., Chisinau, MD-2069, Moldova
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