  Nonlinear dynamical systems may be exposed to tipping points, critical
thresholds at which small changes in the external inputs or in the systems
parameters abruptly shift the system to an alternative state with a contrasting
dynamical behavior. While tipping in a fold bifurcation of an equilibrium is
well understood, much less is known about tipping of oscillations (limit
cycles) though this dynamics are the typical response of many natural systems
to a periodic external forcing, like e.g. seasonal forcing in ecology and
climate sciences. We provide a detailed analysis of tipping phenomena in
periodically forced systems and show that, when limit cycles are considered, a
transient structure, so-called channel, plays a fundamental role in the
transition. Specifically, we demonstrate that trajectories crossing such
channel conserve, for a characteristic time, the twisting behavior of the
stable limit cycle destroyed in the fold bifurcation of cycles. As a
consequence, this channel acts like a ghost of the limit cycle destroyed in the
critical transition and instead of the expected abrupt transition we find a
smooth one. This smoothness is also the reason that it is difficult to
precisely determine the transition point employing the usual indicators of
tipping points, like critical slowing down and flickering.
