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 Şubă ; Silvia Turuta

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2019), pp. 13-40

Résumé

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 $1/l_1^j$).