TY - JOUR

T1 - Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities

AU - Huang, Jianfeng

AU - Liang, Haihua

AU - Llibre, Jaume

PY - 2018/11/5

Y1 - 2018/11/5

N2 - © 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 (cosθ,sinθ)cosθ+Q n (cosθ,sinθ)sinθ,ψ(θ)=Q n (cosθ,sinθ)cosθ−P n (cosθ,sinθ)sinθ,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.

AB - © 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 (cosθ,sinθ)cosθ+Q n (cosθ,sinθ)sinθ,ψ(θ)=Q n (cosθ,sinθ)cosθ−P n (cosθ,sinθ)sinθ,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.

KW - Homogeneous nonlinearities

KW - Limit cycles

KW - Non-existence and uniqueness

KW - Polynomial differential systems

U2 - https://doi.org/10.1016/j.jde.2018.05.019

DO - https://doi.org/10.1016/j.jde.2018.05.019

M3 - Article

SN - 0022-0396

VL - 265

SP - 3888

EP - 3913

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -