Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method

Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method

Year:    2023

Author:    Shijun Liao

Advances in Applied Mathematics and Mechanics, Vol. 15 (2023), Iss. 2 : pp. 267–299

Abstract

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".

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

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

Advances in Applied Mathematics and Mechanics, Vol. 15 (2023), Iss. 2 : pp. 267–299

Published online:    2023-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    33

Keywords:    Small denominator problem Duffing equation limit cycle homotopy analysis method (HAM) MDDiM.

Author Details

Shijun Liao

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