Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method

نویسندگان

چکیده

The so-called ``small denominator problem'' was a fundamental problem of dynamics, as pointed out by Poincar\'{e}. Small denominators appear most commonly in perturbative theory. Duffing equation is the simplest example non-integrable system exhibiting all problems due to small denominators. In this paper, using forced an example, we illustrate that famous problems'' never if non-perturbative approach based on homotopy analysis method (HAM), namely ``the directly defining inverse mapping'' (MDDiM), used. HAM-based MDDiM provides us great freedom define operator undetermined linear so can be completely avoided and besides convergent series multiple limit-cycles with high nonlinearity are successfully obtained. So, from viewpoint HAM, only artifacts perturbation methods. Therefore, abandoning methods but MDDiM, one would troubled denominators''. has general meanings mathematics thus used attack many open related

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ژورنال

عنوان ژورنال: Advances in Applied Mathematics and Mechanics

سال: 2023

ISSN: ['2070-0733', '2075-1354']

DOI: https://doi.org/10.4208/aamm.oa-2022-0260