Czak, Jason Edward2022-05-182022-05-182022-05-17vt_gsexam:34450http://hdl.handle.net/10919/110112In typical attempts to control spatio-temporal chaos, spatially extended systems were subjected to protocols that perturbed them as a whole, often overlooking the potential stabilizing interaction between adjacent regions. We have shown that through the application of a time-delayed feedback scheme to a specific localized region of a system periodic patterns can be generated that are distinct from those observed when controlling the whole system. In this thesis, we present the results of two interconnected studies: 1) Spatio-temporal patterns emerging from spatially localized time-delayed feedback perturbations within transient chaotic states of the Gray-Scott reaction-diffusion system 2) Spatio-temporal patterns emerging from spatially localized time-delayed feedback perturbations within chaotic states of the cubic complex Ginzburg-Landau equation We present an investigation of two model systems: the Gray-Scott reaction-diffusion equation and the complex Ginzburg-Landau equation. Specifically we numerically study two models characterized by exhibiting various chaotic regimes. We first consider a comprehensive study of the Gray-Scott model highlighting key details about different parameter space regimes and their relative proximity to the chaotic regime. Through a systematic investigation of the effects of the model control parameters, time-delayed feedback control strength parameters, perturbed region widths, and other quantities, we show that novel patterns can be formed through the appropriate choice of perturbation region and strength. For the second study we use spatially localized time-delayed feedback on the one-dimensional complex Ginzburg-Landau equation and demonstrate, through the numerical integration of the resulting real and imaginary equations, the stabilization of novel periodic patterns within three distinct chaotic regimes. In these studies we have shown that selectively applying a time-delayed feedback scheme to a specific spatially localized region of a chaotic system can bring forth periodic patterns that are distinct from those observed when applying a perturbation to the whole system. Depending on the protocol used, these new patterns can emerge either in the perturbed or the unperturbed region. The mechanism underlying the observed pattern generation is related to the interplay between diffusion across the interfaces separating the different regions and time-delayed feedback. Research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF-17-1-0156.ETDenIn Copyrightnon-equilibrium systemschaoscontrol theorypattern formationsDissertation