广义Rayleigh-Liénard振子的极限环
The limit cycles in a generalized Rayleigh-Liénard oscillator
摘要：
本文计算了广义Rayleigh-Liénard方程$$\ddot{x}+ax+bx^3-(λ_1+λ_2 x^2+λ_3\dot{x}^2+λ_4 x^4+λ_5\dot{x}^4+λ_6 x^6)\dot{x}=0$$和等价平面系统$X_λ$的开周期环的循环性，其中扰动系数$λ_j$是独立的小参数，$a, b$是固定的非零常数.我们的主要工具是所谓的高阶Poincaré-Pontryagin-Melnikov函数（Melnikov函数，简称$M_n$）的机制，结合中心条件和相应的Bautin理想的显式计算。我们考虑第一任意解析弧$\varepsilon \to λ(\varepsilon)$和明确计算所有可能的Melnikov函数$M_n$有关的变形$X_{ λ(\varepsilon)} $。在第二步，我们获得精确的界限的Melnikov函数（完整的椭圆积分取决于参数）在适当的复杂域中的零点的数量，使用修改的彼得罗夫的方法。为了处理一般情况下的六参数变形$λ\to X_λ$，我们首先计算相关的Bautin理想。为此，我们仔细研究了三阶以下的Melnikov函数，然后使用代数几何中的Nakayama引理。Bautin理想的原理化（在爆破后实现）最终将一般变形$X_{ λ} $的研究减少到单参数变形$X_{ λ(\varepsilon)} $的研究。
英文摘要：
We compute the cyclicity of open period annuli of the following generalized Rayleigh-Liénard equation $$\ddot{x}+ax+bx^3-(λ_1+λ_2 x^2+λ_3\dot{x}^2+λ_4 x^4+λ_5\dot{x}^4+λ_6 x^6)\dot{x}=0$$ and the equivalent planar system $X_λ$, where the coefficients of the perturbation $λ_j$ are independent small parameters and $a, b$ are fixed nonzero constants. Our main tool is the machinery of the so called higher-order Poincaré-Pontryagin-Melnikov functions (Melnikov functions $M_n$ for short), combined with the explicit computation of center conditions and the corresponding Bautin ideal. We consider first arbitrary analytic arcs $\varepsilon \to λ(\varepsilon)$ and explicitly compute all possible Melnikov functions $M_n$ related to the deformation $X_{ λ(\varepsilon)} $. At a second step we obtain exact bounds for the number of the zeros of the Melnikov functions (complete elliptic integrals depending on parameter) in an appropriate complex domain, using a modification of Petrov's method. To deal with the general case of six-parameter deformations $λ\to X_λ$, we compute first the related Bautin ideal. To do this we carefully study the Melnikov functions up to order three, and then use Nakayama lemma from Algebraic geometry. The principalization of the Bautin ideal (achieved after a blow up) reduces finally the study of general deformations $X_{ λ} $ to the study of one-parameter deformations $X_{ λ(\varepsilon)} $.
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