Nonlinear resonance in oscillatory systems with decaying perturbations

TL;DR

This paper investigates how nonlinear oscillatory systems with decaying perturbations exhibit resonance effects, leading to stable phase-locked states, with analysis applied to the Duffing oscillator.

Contribution

It introduces conditions for resonance and stability in nonlinear oscillators with decaying perturbations, combining averaging and stability analysis techniques.

Findings

Resonance effects lead to stable states near periodic solutions.

Conditions for phase-locking regimes are established.

Application to the Duffing oscillator demonstrates practical relevance.

Abstract

Time-decaying perturbations of nonlinear oscillatory systems in the plane are considered. It is assumed that the unperturbed systems are non-isochronous and the perturbations oscillate with an asymptotically constant frequency. Resonance effects and long-term asymptotic regimes for solutions are investigated. In particular, the emergence of stable states close to periodic ones is discussed. By combining the averaging technique and stability analysis, the conditions on perturbations are described that guarantee the existence and stability of the phase-locking regime with a resonant amplitude. The results obtained are applied to the perturbed Duffing oscillator.