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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).

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