摘要
对于一类含单参数Hamilton平面向量场的n次多项式扰动系统(?)，其中λ为小参数，(?)为扰动参数，0<(?)1，k为充分大的正整数，P,Q为实多项式，且degP,degQ≤n，n为非负整数，本文使用坐标变换将此系统化简为含参数的Bogdonov-Takens系统，(?)，其中P~*，Q~*为实多项式，且degP~*，degQ~*≤n，根据系统一阶Mel'nikov函数M_1(h，λ)关于小参数λ的Taylor展开式，直接利用Petrov定理给出了M_1(h，λ)的孤立零点个数上界的估计，即当(?)(m为非负整数)时，B(2，n)≤n+m-1．另外，本文还估算了n=4时扰动系统分岔出极限环个数不超过7，n=3时扰动系统分岔出极限环个数不超过4．
It's studied that the number of isolated zeros of the Mel'nikov function for a one-parameter Hamiltonian system under polynomial perturbations in this paper. The per-turbed system is (?)= -2y + 3λy~2 + (?)P(x, y), (?)= -x - x~2 + (?)Q(x, y), whereλis a small parameter, (?) is a perturbed parameter, (?),k is a positive integer big enough, degP, degQ≤n, and n is a non-negative integer. The system is simplified to a Bogdonov-Takens system with one-parameter by using coordinate transformation, which is (?) = y - 3λy~2 + (?)P~*(x, y), (?)= -x + x~2 + (?)Q~*(x, y), where degP~*, degQ~*≤n. According to the first-order Mel'nikov function M_1(h,λ) in terms of the Taylor expansion about the small parameter A, one upper bound of the number of isolated zeros of M_1 (h,λ) is given by Petrov's theorem. When (?)m/(?)λM_1(h,λ)|_(λ=0) (?)0(m is a non-negative integer), B(2, n)≤n + m-1. Further more, it's obtained that the number of the limit cycles of the perturbed system under quartic polynomial perturbations is no more than 7, and it is no more than 4 under cubic perturbations.
引文
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