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

Jianfeng Huang, Haihua Liang, Jaume Llibre

Research output: Contribution to journalArticleResearch

5 Citations (Scopus)

Abstract

© 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.
Original language English 3888-3913 Journal of Differential Equations 265 https://doi.org/10.1016/j.jde.2018.05.019 Published - 5 Nov 2018

Keywords

• Homogeneous nonlinearities
• Limit cycles
• Non-existence and uniqueness
• Polynomial differential systems