Autoresonance in oscillating systems with combined excitation and weak dissipation

TL;DR

This paper develops a mathematical model to analyze autoresonance in nonlinear oscillating systems with combined excitation and weak dissipation, focusing on solution stability and bifurcations.

Contribution

It introduces a new approach using power-law asymptotics and Lyapunov functions to study autoresonance capture in systems with weak dissipation.

Findings

Existence of unbounded amplitude solutions with phase locking

Stability conditions for autoresonant solutions

Bifurcation analysis under weak dissipation

Abstract

A mathematical model describing the initial stage of the capture into autoresonance for nonlinear oscillating systems with combined parametric and external excitation is considered. The solutions with unboundedly growing amplitude and limited phase mismatch correspond to the autoresonant capture. The paper investigates the existence, stability and bifurcations of such solutions in the presence of a weak dissipation in the system. Our technique is based on the study of particular solutions with power-law asymptotics at infinity and the construction of suitable Lyapunov functions.