Resonances in asymptotically autonomous systems with a decaying chirped-frequency excitation

TL;DR

This paper studies the long-term behavior of strongly nonlinear autonomous systems under decaying, chirped-frequency oscillatory perturbations, revealing conditions for phase locking and drifting, with implications for energy growth and stability.

Contribution

It introduces an analysis combining averaging and Lyapunov methods to characterize asymptotic regimes in systems with time-varying perturbations.

Findings

Identification of phase locking and drifting regimes

Conditions for unbounded energy growth in resonant solutions

Stability analysis of long-term behaviors

Abstract

The influence of oscillatory perturbations on autonomous strongly nonlinear systems in the plane is investigated. It is assumed that the intensity of perturbations decays with time, and their frequency increases according to a power law. The long-term behaviour of perturbed trajectories is discussed. It is shown that, depending on the structure and the parameters of perturbations, there are at least two different asymptotic regimes: a phase locking and a phase drifting. In the case of phase locking, resonant solutions with an unlimitedly growing energy occur. The stability and asymptotics at infinity of such solutions are investigated. The proposed analysis is based on a combination of the averaging technique and the method of Lyapunov functions.