Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities

In this paper we study the limit cycles of the planar polynomial differential systemsx˙=ax−y+Pn(x,y),y˙=x+ay+Qn(x,y), where Pn and Qn are homogeneous polynomials of degree n≥2, and a∈R. Consider the functionsφ(θ)=Pn(cos⁡θ,sin⁡θ)cos⁡θ+Qn(cos⁡θ,sin⁡θ)sin⁡θ,ψ(θ)=Qn(cos⁡θ,sin⁡θ)cos⁡θ−Pn(cos⁡θ,sin⁡θ)sin⁡θ,ω1(θ)=aψ(θ)−φ(θ),ω2(θ)=(n−1)(2aψ(θ)−φ(θ))+ψ′(θ). First we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of ω1 and ω2 with definite sign. This result improves previous known results. Furthermore, if ω1(ν1aψ−ν2φ)≤0 for some ν1,ν2≥0, we provide necessary and sufficient conditions for the non-existence, and the existence and uniqueness of the limit cycles of these differential systems. When one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin.