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

TL;DR

This paper investigates the complex long-term dynamics of a harmonic oscillator subjected to abrupt frequency switches, revealing novel multi-scale behaviors and discrepancies between different modeling approaches.

Contribution

It introduces and analyzes a new class of models for frequency-switching oscillators, uncovering unique aging phenomena and multi-scale oscillations not previously reported.

Findings

Disagreement between two natural models of frequency switching.

Identification of slow-fast staircases and mixed-mode oscillations.

Existence of a synchronized canard explosion proven through asymptotic analysis.

Abstract

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.