Parametric Autoresonance with Time-Delayed Control

TL;DR

This paper explores how a constant time delay affects the stability and control of parametric autoresonance in nonlinear systems, identifying critical thresholds and analytical conditions for sustained resonance.

Contribution

It introduces a novel analysis of delay effects on parametric autoresonance, combining analytical methods with numerical simulations to understand stability thresholds.

Findings

Delay threshold for sustained autoresonance identified

Analytical conditions for phase locking derived

Numerical simulations confirm theoretical predictions

Abstract

We investigate how a constant time delay influences a parametric autoresonant system. This is a nonlinear system driven by a parametrically chirped force with a negative delay-feedback that maintains adiabatic phase locking with the driving frequency. This phase locking results in a continuous amplitude growth, regardless of parameter changes. Our study reveals a critical threshold for delay strength; above this threshold, autoresonance is sustained, while below it, autoresonance diminishes. We examine the interplay between time delay and autoresonance stability, using multi-scale perturbation methods to derive analytical results, which are corroborated by numerical simulations. Ultimately, the goal is to understand and control autoresonance stability through the time-delay parameters.