  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''.
