Phase portraits for quadratic homogeneous polynomial vector fields on S^2

TL;DR

This paper investigates quadratic homogeneous polynomial vector fields on the 2-sphere, providing conditions for singularities, characterizing phase portraits, and analyzing bifurcations, thereby advancing understanding of their global dynamics and limit cycle behavior.

Contribution

It offers necessary and sufficient conditions for singularity types, characterizes global phase portraits, and reduces aspects of Hilbert's 16th problem for this class.

Findings

Homogeneous quadratic vector fields with non-hyperbolic singularities have no limit cycles.

Criteria for identifying centers among singularities on S^2.

Reduction of Hilbert's 16th problem to studying specific families of vector fields.

Abstract

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.