In this paper, we consider polynomial systems of the form x' = y + P(x, y), y' = -x + Q(x, y), where P and Q are polynomials of degree n wihout linear part.
For the case n = 3, we have found new sufficient conditions for a center at the origin, by proposing a first integral linear in certain coefficient of the system. The resulting first integral is in the general case of Darboux type.
By induction, we have been able to generalize these results for polynomial systems of arbitrary degree.
@article{urn:eudml:doc:41244,
title = {Sufficient conditions for the existence of a center in polynomial systems of arbitrary degree.},
journal = {Publicacions Matem\`atiques},
volume = {40},
year = {1996},
pages = {205-214},
mrnumber = {MR1397015},
zbl = {0856.34028},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41244}
}
Giacomini, Hector; Ndiaye, Malick. Sufficient conditions for the existence of a center in polynomial systems of arbitrary degree.. Publicacions Matemàtiques, Tome 40 (1996) pp. 205-214. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41244/