  In the analysis of highly-oscillatory evolution problems, it is commonly
assumed that a single frequency is present and that it is either constant or,
at least, bounded from below by a strictly positive constant uniformly in time.
Allowing for the possibility that the frequency actually depends on time and
vanishes at some instants introduces additional difficulties from both the
asymptotic analysis and numerical simulation points of view. This work is a
first step towards the resolution of these difficulties. In particular, we show
that it is still possible in this situation to infer the asymptotic behaviour
of the solution at the price of more intricate computations and we derive a
second order uniformly accurate numerical method.
