论文标题
通过同型分析方法避免小分母问题
Avoiding small denominator problems by means of the homotopy analysis method
论文作者
论文摘要
如Poincaré所指出的那样，所谓的``小分母问题''是一个基本的动态问题。小分母在扰动理论中最常见。行李方程是由于小分母而表现出所有问题的最简单示例。在本文中，使用强制行驶方程为例，我们说明著名的``小分母问题''永远不会出现，如果使用基于同义分析方法（HAM）的非扰动方法，即``使用``直接定义逆映射''（MDDIM）（MDDIM）的方法。基于HAM的MDDIM为我们提供了极大的自由，可以直接定义不确定的线性算子的反操作员，以便可以完全避免所有小分母，除了收敛的一系列强迫Duff方程的多个极限循环，具有高非线性性能。因此，从火腿的角度来看，著名的``小分母问题''只是扰动方法的伪像。因此，完全放弃了扰动方法，但使用基于HAM的MDDIM，``小分母''永远不会困扰。基于HAM的MDDIM在数学方面具有一般含义，因此可以用来攻击与所谓的``小分母''有关的许多开放问题。
The so-called ``small denominator problem'' was a fundamental problem of dynamics, as pointed out by Poincaré. Small denominators appear most commonly in perturbative theory. The Duffing equation is the simplest example of a non-integrable system exhibiting all problems due to small denominators. In this paper, using the forced Duffing equation as an example, we illustrate that the famous ``small denominator problems'' never appear if a non-perturbative approach based on the homotopy analysis method (HAM), namely ``the method of directly defining inverse mapping'' (MDDiM), is used. The HAM-based MDDiM provides us great freedom to directly define the inverse operator of an undetermined linear operator so that all small denominators can be completely avoided and besides the convergent series of multiple limit-cycles of the forced Duffing equation with high nonlinearity are successfully obtained. So, from the viewpoint of the HAM, the famous ``small denominator problems'' are only artifacts of perturbation methods. Therefore, completely abandoning perturbation methods but using the HAM-based MDDiM, one would be never troubled by ``small denominators''. The HAM-based MDDiM has general meanings in mathematics and thus can be used to attack many open problems related to the so-called ``small denominators''.
