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Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 2019, Number 2, Pages 13–40
(Mi basm505)
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This article is cited in 1 scientific paper (total in 1 paper)
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
Alexandru Şubăa, Silvia Turutab a Vladimir Andrunachievici Institute of Mathematics
and Computer Science,
5 Academiei str., Chişinău, MD 2028, Moldova
b Tiraspol State University,
5 Gh. Iablocichin str., Chişinău, MD-2069, Moldova
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 $m(l_1)+m(Z)\ge 5,$ where $Z=0$ is the line at infinity and $l_1=0$ is an affine real invariant straight line, we prove that the critical point $M_0$ is of the center type if and only if the first Lyapunov quantity vanishes. More over, if $m(Z)=5$ (respectively, $m(l_1)+m(Z)\ge 5,~ m(l_1)\ge j,~ j=2,3 $) then $M_0$ is a center if the cubic systems have a polynomial first integral (respectively, an integrating factor of the form $1/l_1^j$).
Keywords and phrases:
cubic differential system, center problem, invariant straight line, algebraic multiplicity.
Received: 18.03.2019
Citation:
Alexandru Şubă, Silvia Turuta, “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”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2019, no. 2, 13–40
Linking options:
https://www.mathnet.ru/eng/basm505 https://www.mathnet.ru/eng/basm/y2019/i2/p13
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Abstract page: | 238 | Full-text PDF : | 81 | References: | 24 |
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