Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere

Lex X be a homogeneous polynomial vector field of degree n ≥ 3 on S 2 having finitely many invariant circles. Then, for such a vector field X we find upper bounds for the number of invariant circles, invariant great circles, invariant circles intersecting at a same point and parallel circles with the same director vector. We give examples of homogeneous polynomial vector fields of degree 3 on S 2 having finitely many invariant circles which are not great circles, which are limit cycles, but are not great circles and invariant great circles that are limit cycles. Moreover, for the casen=3 we determine the maximum number of parallel invariant circles with the same director vector. © 2006 Springer.