论文标题
广义雷利 - 里纳德振荡器中的极限周期
The limit cycles in a generalized Rayleigh-Liénard oscillator
论文作者
论文摘要
我们计算以下广义雷利 - 莱纳德方程$$ \ ddot {x}+ax+ax+ax+bx^3-（λ_1+λ_2x^2+λ_3\ dot {X} x^6）\ dot {x} = 0 $$和等效的平面系统$x_λ$，其中扰动$λ_j$的系数是独立的小参数，$ a，b $是固定的非零常数。我们的主要工具是所谓的高阶Poincaré-Pontryagin-Melnikov函数（Short Melnikov函数$ M_N $），并结合了中心条件的明确计算和相应的Bautin Ideal。 我们考虑第一个任意分析弧$ \ varepsilon \toλ（\ varepsilon）$，并明确计算与变形$ x_ {λ（\ varepsilon）} $有关的所有可能的Melnikov函数$ M_N $。在第二步中，我们使用PETROV方法的修改，在适当的复杂域中获得了Melnikov函数的零零（完全取决于参数的完整椭圆积分）的精确边界。 为了处理六参数变形的一般情况$λ\至x_λ$，我们首先计算相关的bautin理想。为此，我们仔细研究了Melnikov的功能，直到订购三个，然后使用代数几何形状中的Nakayama引理。黄质理想的原理化（在爆炸后实现）最终减少了对一参数变形的研究$ x_ {λ} $的研究$ x_ {λ（\ varepsilon）} $。
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)} $.
