Pace in Concert with Phase: Rate-induced Phase-tipping in Birhythmic Oscillators

TL;DR

This paper investigates rate-induced phase-tipping in birhythmic oscillators, revealing how parameter change rates and phases trigger abrupt transitions between stable limit cycles, with implications for understanding complex oscillatory behaviors.

Contribution

It introduces the concept of rate-induced phase-tipping in birhythmic oscillators and analyzes its mechanisms through examples like the van der Pol and glycolysis models.

Findings

RP-tipping occurs within specific rate and phase intervals.

Transitions can be irreversible or series of switches.

RP-tipping diagrams map conditions for transitions.

Abstract

We study rate-induced phase-tipping (RP-tipping) between two stable limit cycles of a birhythmic oscillator. We say that such an oscillator RP-tips when a time variation of an input parameter preserves the bistability of the limit cycles but induces transitions from one stable limit cycle to the other, causing abrupt changes in the amplitude and frequency of the oscillations. Crucially, these transitions occur when: the rate of change of the input is in a certain interval bounded by critical rate(s), and the system is in certain phases of the cycle. We focus on two illustrative examples: the birhythmic van der Pol oscillator and the birhythmic Decroly-Goldbeter glycolysis model, each subjected to monotone and non-monotone shifts in their input parameters. We explain RP-tipping in terms of properties of the autonomous frozen system, including the phase of a cycle and partial basin instability along the parameter path traced by the changing input. We show that RP-tipping can occur as an irreversible one-way transition or as a series of transitions between the stable limit cycles. Finally, we present RP-tipping diagrams showing combinations of the rate and magnitude of parameter shifts and the phase of the oscillation that give rise to this genuine non-autonomous instability.