  Bifurcation theory and continuation methods are well-established tools for
the analysis of nonlinear mechanical systems subject to periodic forcing. We
illustrate the added value and the complementary information provided by
singularity theory with one distinguished parameter. While tracking
bifurcations reveals the qualitative changes in the behaviour, tracking
singularities reveals how structural changes are themselves organised in
parameter space. The complementarity of that information is demonstrated in the
analysis of detached resonance curves in a two-degree-of-freedom system.
