Dissipative systems with nonlocal delayed feedback control

TL;DR

This paper analyzes a linear model of a spatially extended dissipative medium under delayed feedback control, revealing how system delay and feedback strength influence stability, bifurcations, and oscillatory behaviors.

Contribution

It introduces a simplified linear model that captures key dynamics of nonlocal delayed feedback control in dissipative systems, including stability criteria and bifurcation phenomena.

Findings

Non-zero system delay destabilizes the fixed point at high feedback strengths.

A supercritical Hopf bifurcation occurs with sigmoid-bounded feedback.

Eigenvalues exhibit discontinuous changes along specific parameter lines.

Abstract

We present a linear model, which mimics the response of a spatially extended dissipative medium to a distant perturbation, and investigate its dynamics under delayed feedback control. The time a perturbation needs to propagate to a measurement point is captured by an inherent delay time (or latency). A detailed linear stability analysis demonstrates that a non-zero system delay acts destabilizing on the otherwise stable fixed point for sufficiently large feedback strengths. The imaginary part of the dominant eigenvalue is bounded by twice the feedback strength. In the relevant parameter space it changes discontinuously along specific lines when switching between branches of eigenvalues. When the feedback control force is bounded by a sigmoid function, a supercritical Hopf bifurcation occurs at the stability-instability transition. The frequency and amplitude of the resulting limit cycles respond to parameter changes like the dominant eigenvalue. In particular, they show similar discontinuities along specific lines. These results are largely independent of the exact shape of the sigmoid function. Our findings match well with previously reported results on a feedback-induced instability of vortex diffusion in a rotationally driven Newtonian fluid [M. Zeitz, P. Gurevich, and H. Stark, Eur. Phys. J. E 38, 22 (2015)]. Thus, our model captures the essential features of nonlocal delayed feedback control in spatially extended dissipative systems.