Periodic forces combined with feedback induce quenching in a bistable oscillator

TL;DR

This paper investigates how periodic forces combined with feedback can induce quenching in bistable oscillators, revealing bifurcation mechanisms and potential control methods for abnormal oscillations.

Contribution

It introduces a state-transition method for quenching oscillations in bistable systems using periodic forces and quadratic feedback, expanding understanding of bifurcation structures.

Findings

Oscillation quenching occurs via homoclinic bifurcation.

Periodic forcing combined with feedback induces amplitude changes.

Rich bifurcation structures are identified in driven bistable oscillators.

Abstract

The coexistence of an abnormal rhythm and a normal steady state is often observed in nature (e.g., epilepsy). Such a system is modeled as a bistable oscillator that possesses both a limit cycle and a fixed point. Although bistable oscillators under several perturbations have been addressed in the literature, the mechanism of oscillation quenching, a transition from a limit cycle to a fixed point, has not been fully understood. In this study, we analyze quenching using the extended Stuart-Landau oscillator driven by periodic forces. Numerical simulations suggest that the entrainment to the periodic force induces the amplitude change of a limit cycle. By reducing the system with an averaging method, we investigate the bifurcation structures of the periodically-driven oscillator. We find that oscillation quenching occurs by the homoclinic bifurcation when we use a periodic force combined with quadratic feedback. In conclusion, we develop a state-transition method in a bistable oscillator using periodic forces, which would have the potential for practical applications in controlling and annihilating abnormal oscillations. Moreover, we clarify the rich and diverse bifurcation structures behind periodically-driven bistable oscillators, which we believe would contribute to further understanding the complex behaviors in non-autonomous systems.