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

© 2018 Elsevier Inc. In this paper we study the limit cycles of the planar polynomial differential systems x˙=ax−y+P n (x,y),y˙=x+ay+Q n (x,y), where P n and Q n are homogeneous polynomials of degree n≥2, and a∈R. Consider the functions φ(θ)=P n (cos⁡θ,sin⁡θ)cos⁡θ+Q n (cos⁡θ,sin⁡θ)sin⁡θ,ψ(θ)=Q n (cos⁡θ,sin⁡θ)cos⁡θ−P n (cos⁡θ,sin⁡θ)sin⁡θ,ω 1 (θ)=aψ(θ)−φ(θ),ω 2 (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 ω 1 and ω 2 with definite sign. This result improves previous known results. Furthermore, if ω 1 (ν 1 aψ−ν 2 φ)≤0 for some ν 1 ,ν 2 ≥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.