Let X be a homogeneous polynomial vector field of degree 2 on S 2. We show that if X has at least a non-hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on S 2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 th Hilbert's problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on S 2 of degree n. © 2009 Springer-Verlag Italia.
|Journal||Rendiconti del Circolo Matematico di Palermo|
|Publication status||Published - 1 Dec 2009|
- Invariant algebraic curves
- Limit cycles
- Polynomial vector field