Volume 15, Issue 2
Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method

Adv. Appl. Math. Mech., 15 (2023), pp. 267-299.

Published online: 2022-12

[An open-access article; the PDF is free to any online user.]

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• 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".

41A58, 34C25

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@Article{AAMM-15-267, author = {Liao , Shijun}, title = {Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {15}, number = {2}, pages = {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".

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2022-0260}, url = {http://global-sci.org/intro/article_detail/aamm/21269.html} }
TY - JOUR T1 - Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method AU - Liao , Shijun JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 267 EP - 299 PY - 2022 DA - 2022/12 SN - 15 DO - http://doi.org/10.4208/aamm.OA-2022-0260 UR - https://global-sci.org/intro/article_detail/aamm/21269.html KW - Small denominator problem, Duffing equation, limit cycle, homotopy analysis method (HAM), MDDiM. AB -

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".

Shijun Liao. (2022). Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method. Advances in Applied Mathematics and Mechanics. 15 (2). 267-299. doi:10.4208/aamm.OA-2022-0260
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