Time-Delay Feedback Methods and Their Application In Oscillatory Systems

Abstract

Reaction-diffusion systems are a paradigm for the study of nonlinear dynamical processes that are taking place in a spatially-extended medium. Examples can be found in many natural systems, ranging from physics and chemistry to biology, and are also applied in areas as different as finances and cultural anthropology. Common aspects of these systems are that they show coherent temporal and spatiotemporal behaviour, reflected by travelling wave solutions, uniform oscillations, spatially periodic but temporally constant patterns (like the famous Turing patterns), or localised patterns like spots. Over the years, the focus of the research of these systems has moved towards the controlling and self-engineering of patterns and systems, notably to the inclusion of feedback loops, designed for stabilising spatiotemporal chaos or inducing novel patterns. These feedback loops can be a consequence of the system dynamics itself (are intrinsic) and are often operating with a time delay since assuming an instantaneous feedback is unrealistic in many cases. Specifically, in oscillatory reaction-diffusion systems, spatial coupling can render uniform oscillations unstable and can lead to spatiotemporal chaos. The application of time-delay terms then can stabilize a range of different, including novel, regular solutions. In this thesis, the complex Ginzburg- Landau equation in one-dimensional space subjected to a combined global and local time delayed feedback term has been studied and investigated (a) travelling waves and (b) localized spot patterns. While the travelling wave pattern was found to be transient in simulations, the spot patterns were stable. These patterns are characterized by a change of oscillation amplitude and constant phase shift between the background oscillations and the inside of the localized pattern. The stability area in parameter space, the spatial extension of spots as function of the feedback parameters and the main instabilities has been investigated.