  We study the Duffing equation and its generalizations with polynomial
nonlinearities. Recently, we have demonstrated that metamorphoses of the
amplitude response curves, computed by asymptotic methods in implicit form as
$F\left( \Omega ,\ A\right) =0$, permit prediction of qualitative changes of
dynamics occurring at singular points of the implicit curve $F\left(\Omega ,\
A\right) =0$. In the present work we determine a global structure of singular
points of the amplitude profiles computing bifurcation sets, i.e. sets
containing all points in the parameter space for which the amplitude profile
has a singular point. We connect our work with independent research on
tangential points on amplitude profiles, associated with jump phenomena,
characteristic for the Duffing equation. We also show that our techniques can
be applied to solutions of form $\Omega _{\pm }=f_{\pm }\left( A\right) $,
obtained within other asymptotic approaches.
