Avoiding small denominator problems by means of the homotopy analysis method

TL;DR

This paper demonstrates that the homotopy analysis method (HAM) with the method of directly defining inverse mapping (MDDiM) can completely avoid small denominator problems in nonlinear dynamics, providing convergent solutions for the forced Duffing equation.

Contribution

The paper introduces a non-perturbative HAM-based MDDiM approach that avoids small denominators, unlike traditional perturbation methods, and successfully computes multiple limit-cycles.

Findings

Small denominator problems are artifacts of perturbation methods.

HAM-based MDDiM avoids small denominators in nonlinear problems.

Convergent series solutions for the forced Duffing equation are obtained.

Abstract

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. 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''.