Novel slow–fast behaviour in an oscillator driven by a frequency-switching force

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