余弦广义Padé逼近法及其在强非线性振子周期解求解中的应用

Application of the consine generalized Padé approximation method in solving periodic solutions of strongly nonlinear oscillators

基于广义Padé逼近方法,构造了一类余弦型广义Padé逼近式,并针对不同类型振子周期轨道的特性,对广义Padé逼近法的求解过程进行了改进。基于改进后的方法求得了一类势能函数为高阶多项式、有理函数和无理函数振子的解析近似周期解。通过与数值解进行比较,验证了所得之解有着较高的精度和可靠性,且不受非线性项系数大小和初始振幅大小的影响。同时,该方法也不局限于某个特定的系统,而是具有较广的适用范围。上述结果说明,通过合理构造广义Padé逼近式,Padé逼近方法亦可直接用于周期解的求解,为Padé逼近在振动领域中的应用提供了新的思路和参考方法。

Based on the generalized Padéapproximate method,a cosine type generalized Padéapproximation was constructed.According to the traits of different oscillators,the method was further modified,via which the periodic solutions of a kind of strongly nonlinear autonomous oscillators with its potential function expressed as a high order polynomial function,rational function or irrational function were obtained.Compared with numerical solutions,the precision and reliability of the proposed method were proved.In addition,the precision of the solutions keeps high in despite of that the nonlinear parameters or initial amplitude are large or small.Besides,the proposed method can be utilized in many kinds of systems,which means that the proposed method is generally applicable in wide ranges.The results show that the Padéapproximate method can be utilized to solve periodic solutions directly by constructing appropriate generalized Padéappromate terms and can also provide some new considerations and reference methods.