Dissipative systems with nonlocal delayed feedback control

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 nonzero system delay acts to destabilize 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 eigenvalues. When the feedback control force is bounded by a sigmoid function, a supercritical Hopf bifurcation occurs at the stability-instability transition. Perturbing the fixed point, 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 (Zeitz et al 2015 Eur. Phys. J. E 38 22). Thus, our model captures the essential features of nonlocal delayed feedback control in spatially extended dissipative systems.

Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay

“…From a practical point of view, delay is often unwanted, as it may destroy the system's stability [32][33][34][35][36][37][38] or induce chaos [39,40]. It furthermore dramatically increases the difficulty of the mathematical description, as the combination of delay and noise yields non-Markovian stochastic processes with infinite-dimensionality [31,32,37,38,41], that are, moreover, automatically far from thermal equilibrium [42,43].…”