Abstract
The authors report the emergence of antiresonance in a variation of the Stuart-Landau model, which alternates between two unstable modes but displays a stable response in a finite range of switching frequencies.
Highlights
Switched dynamics are pervasive in theoretical physics, neuroscience, and engineering [1]
Antiresonance of switched systems with only unstable modes has never been documented. Is it possible to induce antiresonance in the form of stable response of a system switching between two unstable modes? Here, we provide a positive answer to this question by offering the first example of a switched system composed of only unstable modes that displays a stable response in a finite range of switching frequencies
The main differences with respect to the present analysis are: (i) our analysis is for a general value of δ, while the setups in Refs. [23,24] assume equal probability for the on and off states, (ii) we constructively demonstrate a compact window of stability for the switching rate and provide a closed form result for it, while Ref. [24] identifies integer values that are pertinent to discrete dynamics and Ref. [23] only points at the existence of a rate that guarantees instability, and (iii) we address the dependence on initial conditions toward the characterization of the basin of attraction of the original switched system, which is not part of the analysis in Refs. [23,24]
Summary
Russell Jeter Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA. Igor Belykh † Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA and Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, 603950 Nizhny Novgorod, Russia (Received 9 October 2020; revised 2 December 2020; accepted 23 February 2021; published 1 April 2021). Antiresonance is a key property of dynamical systems that leads to the suppression of oscillations at select frequencies. We elucidate the stabilization mechanism and characterize the range of antiresonant frequencies for periodic and stochastic switching. The demonstration of antiresonance in a minimalistic variation of the Stuart-Landau model opens the door for a new paradigm in the study and design of switched systems
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