  Geometric singular perturbation theory provides a powerful mathematical
framework for the analysis of 'stationary' multiple time-scale systems which
possess a critical manifold, i.e. a smooth manifold of steady states for the
limiting fast subsystem, particularly when combined with a method of
desingularization known as blow-up. The theory for 'oscillatory' multiple
time-scale systems which possess a limit cycle manifold instead of (or in
addition to) a critical manifold is less developed, particularly in the
non-normally hyperbolic regime. We use the blow-up method to analyse the global
oscillatory transition near a regular folded limit cycle manifold in a class of
three time-scale 'semi-oscillatory' systems with two small parameters. The
systems considered behave like oscillatory systems as the smallest perturbation
parameter tends to zero, and stationary systems as both perturbation parameters
tend to zero. The additional time-scale structure is crucial for the
applicability of the blow-up method, which cannot be applied directly to the
two time-scale oscillatory counterpart of the problem. Our methods allow us to
describe the asymptotics and strong contractivity of all solutions which
traverse a neighbourhood of the global singularity. Our main results cover a
range of different cases with respect to the relative time-scale of the angular
dynamics and the parameter drift. We demonstrate the applicability of our
results for systems with periodic forcing in the slow equation, in particular
for a class of Li\'enard equations. Finally, we consider a toy model used to
study tipping phenomena in climate systems with periodic forcing in the fast
equation, which violates the conditions of our main results, in order to
demonstrate the applicability of classical (two time-scale) theory for problems
of this kind.
