Resonance in isochronous systems with decaying oscillatory perturbations

TL;DR

This paper investigates how decaying oscillatory perturbations affect isochronous systems, leading to the emergence of stable resonant solutions with constant amplitude under certain conditions.

Contribution

It introduces a combined averaging and Lyapunov approach to analyze the existence and stability of resonant solutions in perturbed isochronous systems.

Findings

Resonant solutions can be attracting with constant amplitude

Conditions for phase locking and drifting regimes are identified

Stability criteria for resonant dynamics are established

Abstract

Non-autonomous perturbations of isochronous systems in the plane are considered. It is assumed that the intensity of perturbations decays with time, and the frequency is asymptotically constant with the limiting value satisfying a resonance condition. We discuss the emergence of attracting resonant solutions with an asymptotically constant amplitude. By combining the averaging technique and the Lyapunov function method, we show that this behaviour can occur in the phase locking and phase drifting regimes. The conditions that guarantee the existence and stability of such resonant dynamics are described.