Parametric Autoresonance with Time-Delayed Control

TL;DR

This work addresses controlling parametric autoresonance in a nonlinear oscillator with time-delayed negative feedback. The authors formulate a model with a chirped parametric drive and derive slow-flow equations using a multi-scale perturbation approach, obtaining a fixed-point condition that yields a detuning relation $\sigma$ and an explicit quasi-steady amplitude $\hat{A}$. A delay-threshold $k_{th}=\frac{\gamma\omega}{\sin(\omega\bar{\beta})}$ is derived from stability analysis and validated by numerical simulations, showing autoresonance for $k>k_{th}$ and decay for $k<k_{th}$, with the threshold largely independent of the forcing amplitude $h$ but interacting with the delay $\bar{\beta}$. The results provide a delay-based control mechanism for sustaining resonance with potential applications in mechanical, electrical, and fluid systems.

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.

Figures (6)

• Figure 1: Analytical plot of amplitude growth over time using Eq. (13) for parameters $\gamma = 0.002$, $\omega = 0.5$, and $\alpha = 0.08$. Threshold values: (a) $k_{\text{th}} = 0.02$ for $\bar{\beta} = 0.1$ and (b) $k_{\text{th}} = 0.0066$ for $\bar{\beta} = 0.3$ indicate the onset of autoresonant growth when crossed.
• Figure 2: Panels (a) and (b) show the oscillations of the system with their maximum envelope. Panels (c) and (d) show the phase space orbits. Panels (e) and (f) show the maximum of the oscillations and the fit curve of the asymptotic part. The parameters are $h=0.002,~\Bar{\beta}=0.0092$, and $k=0.02$ in the panels in the left column, while in the panels in the right column $k=0.024$. In panels (e) and (f) we show the maximum envelope of the oscillations with the fitted curve and its equation.
• Figure 3: The figure shows the maximum and minimum diagram, the oscillations amplitude, and the exponent $b$ of the fitting of the oscillations maximum versus the parameter $k$, for $\Bar{\beta}=0.1$ and $h=0.001$ in the panels in the left column and for $h=0.002$ in the panels in the right column. As expected from the analytical analysis the autoresonance starts, in both cases, for $k = 0.0201 \simeq k_{th}$, which is indicated by the vertical red lines in panels (c)-(f). The horizontal red lines in figures (e) and (f) indicate $b=0$, which marks the point where the value of $b$ changes from negative to positive, signifying the onset of the autoresonance.
• Figure 4: Figures (a) and (b) show the maximum and minimum diagram, figures (c) and (d) the oscillations amplitude, and figures (e) and (f) the exponent $b$ of the exponential fitting curve of the oscillations maximum versus the parameter $\Bar{\beta}$, for $h=0.002$. We have fixed $k=0.02$ for the panels in the left column and $k=0.024$ for the panels in the right column. The horizontal red lines mark the zero to underline the change from negative to positive of the $b$ value, which indicates that the autoresonance is taking place.
• Figure 5: The figure shows the theoretical dependence of $\Bar{\beta}_{th}$ as a function of $k$ as resolved in Eq \ref{['eq:taut']}. The red and green lines illustrate the cases already exposed in Fig. \ref{['fig:0']}, $\Bar{\beta}=0.1$ and $k=0.02$ and $\Bar{\beta}=0.3$ and $k=0.0066$, respectively.