以外的特定区域,覆盖在这项研究中,研究不包括内部、外部和复合的扩张通过共振的角度进行识别。高维问题的言论也没有在本研究发现有效的情况下应用于多个角度理解。此外,该研究还针对二阶和高阶平均不覆盖阶段之间的平均正常形式(阿罗史密斯&的地方,1990年,p.352). .这项研究仅局限于平均,不包含在它的几个经典,第一级和正常形式。将推荐透露一些具体发现非线性动力系统的渐近分析领域未来的研究可以帮助在更高级别范式理论和in-variant流形的平均使用相关定理和示例(p.88桑德斯et al .,2007). .此外,未来研究的重要方法中未涉及本研究即平均偏微分方程的方法也会推荐。此外,天体力学的研究也被排除在本研究(p.88桑德斯et al .,2007)。这个建议,因为意义给了天体力学的勘探异常丰富的资源来找出数学难题的解决方案。因此,这些地区可以有效地描述是用来强调数学和理论解释的不同方面的进一步影响非线性动力系统的渐近分析(p.352阿罗史密斯&,1990)。
Other than the specific areas that are covered in this study, the research does not cover the inner, outer and composite expansion of the angles for identification of the passage through resonance. The remarks on higher dimensional problems are also not covered in this research yet they are found effective in understanding the cases applied on more than one angle. In addition, the research is also specific to second order and higher order averaging not covering the stages between the averaging to normal forms (Arrowsmith & Place, 1990, p.352).. The study is restricted to the averaging only and not encompassing under it the several classical, first level and normal forms. It would be recommended to reveal some specific uncovered areas of an asymptotic analysis of nonlinear dynamical system in future studies that can help in covering the higher level normal form theories and in-variant manifolds of averaging using relevant theorems and examples (Sanders et al., 2007, p.88).. Furthermore, a future study on the significant method not covered in this research i.e. averaging methods for partial differential equations would also be recommended. In addition, study of the celestial mechanics has also been excluded from this research (Sanders et al., 2007, p.88). This recommendation is made because of the significance given to the celestial mechanics for the exploration of the exceptional rich sources to find out the solutions of the difficult problems in mathematics. Hence, these described areas can effectively be used to highlight different aspects of the mathematical and theoretical explanation of the further implications of the asymptotic analysis of nonlinear dynamical system (Arrowsmith & Place, 1990, p.352).