Resonance in Isochronous Systems with Decaying Oscillatory Perturbations

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.