高频扰动下同宿轨的Melnikov函数的估计

Estimation of Melnikov Function for Homoclinic Orbit under High Frequency Perturbations

讨论了高频扰动下同宿轨的Melnikov函数,以Fourier变换为工具,证明了对一般的周期扰动,如果扰动项关于空间变量是Ck(k≥1)的,则当扰动频率ω→∞时,相应的Melnikov函数是■(1/ωk+1)的.

In this paper we discuss the Melnikov function for homoclinic orbit under high frequency perturbations. By using the method of Fourier transformation, we prove that for a general periodic perturbation, if the perturbation term is C^k spatially, then the Melnikov function is （ 1/ω^k＋1） as ω→∞ .