A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities
A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities
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齐次非线性平面多项式系统极限环唯一性判据
DOI：
10.1016/j.jmaa.2017.08.008
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发表时间：
2018
影响因子：
1.3
通讯作者：
Liang Haihua
中科院分区：
文献类型：
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作者：
Huang Jianfeng;Liang Haihua
研究了平面多项式系统：xstec = ax-y + Pn（x，y），ystec = x+ ay + Qn（x，y），其中a∈ R，Pn，Qn是n次≥ 2的齐次多项式.记为n（θ）= cos n（θ）<$Q n（cos n（θ），sin n（θ））− sin n（θ）<$P n（cos n（θ），sin n（θ））。我们证明了当（n− 1）a θ（θ）+ θ stec（θ）<$0时，系统至多有1个围绕原点的极限环.此外，这个上界是尖锐的。这可能是此类系统极限环的第一个唯一性准则，该准则仅取决于ψ的（线性）条件。我们通过例子表明，在许多情况下，该准则是适用的，而经典的是无效的。本文主要使用的工具是系数为不定号的Abel方程极限环个数的一个新的估计。利用这个工具，我们还得到了另一个几何判据，它允许系统在原点周围最多有2个极限环。
This paper is devoted to study the planar polynomial system: x˙= a x− y+ P n (x, y), y˙= x+ a y+ Q n (x, y), where a∈ R and P n, Q n are homogeneous polynomials of degree n≥ 2. Denote ψ (θ)= cos(θ)⋅ Q n (cos(θ), sin(θ))− sin(θ)⋅ P n (cos(θ), sin(θ)). We prove that the system has at most 1 limit cycle surrounding the origin provided (n− 1) a ψ (θ)+ ψ˙(θ)≠ 0. Furthermore, this upper bound is sharp. This is maybe the first uniqueness criterion, which only depends on a (linear) condition of ψ, for the limit cycles of this kind of systems. We show by examples that in many cases, the criterion is applicable while the classical ones are invalid. The tool that we mainly use is a new estimate for the number of limit cycles of Abel equation with coefficients of indefinite signs. Employing this tool, we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin.