Pattern transitions induced by delay feedback

Abstract

Modulated by delay feedback (DF), a reaction-diffusion system is destabilized and undergoes pattern transitions in the parametric region where the undelayed system spontaneously exhibits a bulk oscillation. By varying the feedback parameters, oscillatory hexagon superlattices and stripes, as well as stationary hexagons are observed. Meanwhile, the hexagon superlattices with different wavelengths are found under appropriate feedback parameters. It is demonstrated that, since the DF induces an instability of homogeneous limit cycle with respect to spatial perturbations, the patterns possessing the corresponding spatial modes are formed. Instead of stabilizing the system, here the DF may play a role of destabilization.

Stationary localized structures and the effect of the delayed feedback in the Brusselator model

“…In driven Kerr cavities described by the Lugiato-Lefever equation, timedelayed feedback induces a drift of localized structures, and the route to spatiotemporal chaos has been discussed in [ Recently, time-delayed feedback control has attracted a lot of interest in various fields of nonlinear science such as nonlinear optics, fibre optics, biology, ecology, fluid mechanics, granular matter, plant ecology (see recent overview [67]), and the excellent book by Erneux [54]. Numerical simulations of the Brusselator model with delayed feedback using Pyragas control have provided evidence of moving periodic structures in the form of stripes and hexagons [68] or superlattices [69]. Dissipative structures and coherence resonance in the stochastic Swift-Hohenberg equation with Pyragas control have also been investigated [70].…”

The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

“…In [19], Ott et al first proposed delay to control system by utilizing the input signals adjusted to the temporal states of the system, and then delayed feedback and its modifications are widely applied to control chaos and to stabilize unstable oscillations. Motivated by the idea of Ott and Grebogi, many investigators have studied the effect of time delay in ecological and chemical models (see [1,2,7,9,13,16,24,28,30]). Although many researches have been devoted to the experiments about the suppression of the delayed feedback on the chemical turbulent, there is few analysis of the effect of delayed feedback on the dynamics of chemical reaction models theoretically.…”

“…Numerical simulations of system(2) under the data (D1), where τ = 1 > τ * ≈ 0.7834. The positive equilibrium E * (1.0847, 0.8500) of system (2) becomes unstable, and the bifurcating periodic solutions from E * is stable.From(16) we have τ + 0,j ≈ 0.7834 + 10.0194j for j ∈ N 0 .…”

Effects of stochastic time-delayed feedback on a dynamical system modeling a chemical oscillator

“…Control of spatiotemporal chaotic patterns has been attained in the reaction-diffusion Gray-Scott two-species model by means of time-delayed feedback [31]. This strategy can be harnessed to induce pattern transitions [32], spiral waves and their modulation [33], and complex dynamics of localized structures [34] and Turing patterns [35] in reaction diffusion systems. The emergence of spatial patterns triggered by a time delay via a Hopf bifurcation is observed in a plankton prey-predator model [36].…”