  When an oscillator switches abruptly between different frequencies, there is
some ambiguity in deciding how the system should be modelled at the switch.
Here we describe two seemingly natural models of a switch in a simple
periodically-forced harmonic oscillator, which disagree starkly in their
predictions of its long time behaviour. Attempting to resolve the disagreement
by `regularizing' the switch not only preserves the disagreement, but shows it
increases with time. One of the models corresponds to a conventional `Filippov'
description of a nonsmooth system, while the second exhibits a structure that
irreversibly ages, developing a number of novel multi-scale behaviours that we
believe have not been reported before. These include slow-fast staircases,
novel mixed-mode oscillations, and a synchronized canard explosion. These
features are proven to exist using asymptotic analysis, but as they involve a
slow-fast time-scale separation that increases with time, they lie beyond the
reach of numerical methods.
