Let X be a homogeneous polynomial vector field of degree 2 on S2 having finitely many invariant circles. Then, we prove that each invariant circle is a great circle of S2, and at most there are two invariant circles. We characterize the global phase portrait of these vector fields. Moreover, we show that if X has at least an invariant circle then it does not have limit cycles.
@article{urn:eudml:doc:41857,
title = {Homogeneous polynomial vector fields of degree 2 on the 2-dimensional sphere.},
journal = {Extracta Mathematicae},
volume = {21},
year = {2006},
pages = {167-190},
zbl = {1129.34025},
mrnumber = {MR2292746},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41857}
}
Llibre, Jaume; Pessoa, Claudio. Homogeneous polynomial vector fields of degree 2 on the 2-dimensional sphere.. Extracta Mathematicae, Tome 21 (2006) pp. 167-190. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41857/