Stabilisation of dynamics of oscillatory systems by non-autonomous perturbation

TL;DR

This paper investigates how slowly varying, aperiodic driving forces can enhance the stability of oscillatory systems, expanding the parameter space for synchronization and demonstrating a stabilizing effect in both simple and complex oscillators.

Contribution

It reveals that non-autonomous, aperiodic perturbations can enlarge the Arnold tongue and stabilize complex dynamics, a novel insight into oscillator stability under realistic driving conditions.

Findings

Slowly varying frequency enlarges the Arnold tongue.

Intermittent synchronization leads to average stability.

Numerical evidence of stabilization in higher-dimensional oscillators.

Abstract

Synchronisation and stability under periodic oscillatory driving are well-understood, but little is known about the effects of aperiodic driving, despite its abundance in nature. Here, we consider oscillators subject to driving with slowly varying frequency, and investigate both short-term and long-term stability properties. For a phase oscillator, we find that, counter-intuitively, such variation is guaranteed to enlarge the Arnold tongue in parameter space. Using analytical and numerical methods that provide information on time-variable dynamical properties, we find that the growth of the Arnold tongue is specifically due to the growth of a region of intermittent synchronisation where trajectories alternate between short-term stability and short-term neutral stability, giving rise to stability on average. We also present examples of higher-dimensional nonlinear oscillators where a similar stabilisation phenomenon is numerically observed. Our findings help support the case that in general, deterministic non-autonomous perturbation is a very good candidate for stabilising complex dynamics.