Asymptotic analysis of the autoresonance capture in oscillating systems with combined excitation

TL;DR

This paper analyzes the conditions for stable autoresonance capture in nonlinear oscillating systems with combined parametric and external chirped excitation, using Lyapunov methods to derive asymptotic stability criteria.

Contribution

It provides a mathematical framework for understanding the stability thresholds of autoresonant modes in systems with combined excitation.

Findings

Stable autoresonant solutions exist under specific parameter conditions.

Unstable regimes can be stabilized when parameters cross certain thresholds.

Asymptotic formulas describe long-term behavior of stable solutions.

Abstract

A mathematical model of autoresonance in nonlinear systems with combined parametric and external chirped frequency excitation is considered. Solutions with a growing amplitude and a bounded phase mismatch are associated with the autoresonant capture. By applying Lyapunov function method we investigate the conditions for the existence and stability of autoresonant modes and construct long-term asymptotics for stable solutions. In particular, we show that unstable regimes become stable when the system parameters pass through certain threshold values.