# Bifurcations of autoresonant modes in oscillating systems with combined excitation

## TL;DR

This paper models how nonlinear oscillating systems can be captured into autoresonance through combined parametric and external periodic perturbations, analyzing stability and bifurcations of autoresonant modes over long times.

## Contribution

It introduces a mathematical framework for analyzing bifurcations and stability of autoresonant modes under combined excitation, using a modified averaging method and Lyapunov functions.

## Key findings

- Autoresonant solutions exhibit bifurcations as parameters vary.
- Stability of autoresonant modes depends on bifurcation parameters.
- Asymptotic approximations effectively describe long-term autoresonant behavior.

## Abstract

A mathematical model describing the capture of nonlinear systems into the autoresonance by a combined parametric and external periodic slowly varying perturbation is considered. The autoresonance phenomenon is associated with solutions having an unboundedly growing amplitude and a limited phase mismatch. The paper investigates the behaviour of such solutions when the parameters of the excitation pass through bifurcation values. In particular, the stability of different autoresonant modes is analyzed and the asymptotic approximations of autoresonant solutions on asymptotically long time intervals are proposed by a modified averaging method with using the constructed Lyapunov functions.