In 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. / Doctor of Philosophy / In 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. We present an investigation of two model systems: the Gray-Scott reaction-diffusion equation and the complex Ginzburg-Landau equation. 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 chaotic regimes. In these studies we have shown that selectively applying a time-delayed feedback scheme to a specific region of a chaotic system can generate periodic patterns that are distinct from those observed when controlling 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.

We present an investigation of two non-equilibrium systems: spatial many-species predator-prey games and systems of interacting magnetic skyrmions. We numerically study two predator-prey systems characterized by nested pattern formations. We first consider a six species game in which spiral patterns spontaneously form within coarsening domains. Through a systematic investigation of relevant correlation functions, the interface width, and other quantities, we show that the non-trivial in-domain dynamics affect the coarsening process and the interfacial properties. The exponents which govern domain growth, aging, and interface fluctuations differ from those expected from curvature driven coarsening. The response to perturbations of the reaction rates is also studied. Furthermore, we introduce a nine species model characterized by nested spiral pattern formations. Quantitative evidence of the existence of two length and time scales associated to the spiral levels is presented in the form of correlation lengths and a temporal Fourier analysis of the species densities. A generalized interaction scheme is proposed for dynamically generated hierarchies. Magnetic skyrmions are particle-like spin configurations found in certain chiral magnets. We study the effect of the Magnus force on the relaxation dynamics through Langevin molecular dynamics simulations. The Magnus force enhances the disorder of the system at high noise strengths while we observe a dynamic regime with slow decaying correlations at low noise strengths. The different regimes are characterized by changes in the aging exponent. In general, the Magnus force accelerates the approach to the steady state. In the presence of quenched disorder, we find that the relaxation dynamics are more robust in systems with a strong Magnus force. We also examine periodically driven skyrmion systems and show that a transition from reversible to irreversible flow exists in the presence of attractive defects. The Magnus force enhances the irreversible regime in this case. The work on predator-prey systems was supported by the U.S. National Science Foundation through Grant No. DMR-1606814 whereas the work on skyrmions was supported by the US Department of Energy, Office of Basic Energy Sciences (DOE-BES), under Grant No. DE-FG02-09ER46613. / Doctor of Philosophy / We present an investigation of two non-equilibrium systems: spatial many-species predator- prey games and systems of interacting magnetic skyrmions. We numerically study two predator-prey systems characterized by nested pattern formations. We first consider a six species game in which spiral patterns spontaneously form within coarsening domains. Through a systematic investigation of relevant correlation functions, the interface width, and other quantities, we show that the non-trivial in-domain dynamics affect the coarsening process and, to a greater extent, properties at the interface between competing groups of species. The exponents which govern domain growth, aging, and interface fluctuations are shown to differ from those expected in typical games of competition. We also study the change of the system due to a perturbation of the reaction rates, which could represent an abrupt change in the environment. Furthermore, we introduce a nine species model characterized by the emergence of nested spiral pattern formations. Quantitative evidence of the existence of two distinct spiral levels is presented. We also propose a generalized interaction scheme for dynamically generated spiral hierarchies. Magnetic skyrmions are particle-like spin configurations found in certain chiral magnets. We study the effect of the Magnus force on the dynamic properties of skyrmion systems through particle-based simulations. The Magnus force enhances the disorder of the system at high noise strengths while accelerating the formation of the triangular lattice at low noise strengths. We find that, in general, the Magnus force accelerates the approach to the steady state. In the presence of randomly placed attractive pinning sites, we find that a strong Magnus force can prevent caging effects and allow skyrmions to more easily move around pinning sites. We also examine periodically driven skyrmion systems and show that a transition from reversible to irreversible flow exists in the presence of attractive defects. The Magnus force is shown to enhance the irreversible regime in this case. The work on predator-prey systems was supported by the U.S. National Science Foundation through Grant No. DMR-1606814 whereas the work on skyrmions was supported by the US Department of Energy, Office of Basic Energy Sciences (DOE-BES), under Grant No. DE-FG02-09ER46613.