The background material that is comprised of the existence and uniqueness theorem for initial value problems that are based on contraction and interlinked with this continuation outcomes and growth estimation is helpful in analyzing the basic concepts (Sanders et al., 2007, p.43). The general forms of the equation that is used in the dissertation are the x=f(x,t,) where x and f(x,t,) are vectors elements of real numbers. The variable‘t’ belongs to the real number that is usually associated with the time. However, ‘’ plays a significant part as the small parameter that characterizes the magnitude of the particular perturbations in the differential equations (Sanders et al., 2007, p.43).
The method averaging is the powerful tool of investigation of several perturbation issues in non-linear oscillations and some celestial mechanics (Jungel, 2004, p.86). Moreover, in order to understand the significant influence of small perturbation on the ordinary differential equations with the oscillatory solutions over the long period several similar asymptotic techniques have been created. The method of averaging has a strong history that relies on the transformation to the standard form and then the elimination of the secular terms by employing a near identity transformation (Sanders et al., 2007, p.65).
背景材料是由唯一性定理的存在为初始值问题，基于收缩和延续的结果和增长估计相通的是分析的基本概念有帮助（Sanders等人。，2007，第43页）。这是用在论文的方程的一般形式是X = F（X，T，ε）在X和F（X，T，ε）是实数向量元素。变量不属于真正的数量通常与时间有关的。然而，“ε”中扮演一个重要的部分，作为小参数，表征特定的扰动的幅度在微分方程（Sanders等人。，2007，第43页）。
该方法在非线性振动的平均数摄动问题的调查的有力工具和一些天体力学（云格尔，2004，86页）。此外，为了了解小扰动与长期几个类似的渐近技术的振动解常微分方程的显着的影响已创建。平均的方法，具有很强的历史，依赖于标准形式的转变，然后利用附近的一个恒等变换消除世俗的条款（Sanders等人。，2007，节）。
