Stabilization of Periodic Regimes in Symmetric Systems with Memory via Time-Delayed Feedback Control

University Of Texas At Dallas, Richardson TX

Investigators

Abstract

The symmetric character of fundamental laws of nature manifests itself in an abundance of symmetries in natural phenomena. Modeling of such phenomena by networks of interacting oscillators is common in various fields of science and technology ranging from electronics and optical communications to neurology and medicine. Controlling symmetric features of spontaneously generated oscillatory patterns in such networks constitutes a substantial challenge. The goal of this project is to develop new methods that allow simple but effective control of performance of complex symmetric networks of oscillators. The work is focused on non-invasive methods of control that allow achievement of the desired behavior with minimal forcing of the system and that can be easily adopted for practical implementations (e.g. in electronic circuits). The educational component of this project aims to allow students from the undergraduate to the post-graduate levels to actively participate in this research, working on concrete networks of oscillators with specific symmetry groups. Stabilization of unstable systems is one of the most important problems in applied nonlinear science. Natural symmetries of real life systems lead to multiple periodic regimes and complicate the stabilization problem. The objective of this research is to develop methods for stabilization of unstable periodic regimes in symmetric systems by a combination of delayed feedback control and linear proportional control. The focus of the project will be on two types of symmetric systems with memory, namely systems with hysteresis and with delay. The method of stabilization of unstable periodic orbits (UPO) by non-invasive delayed feedback control (due to Pyragas) has been successfully tested against different types of models of chaotic systems. During the last two years, the co-PIs have been developing computerized symbolic numerical tools and supporting theoretical framework for localization and classification of UPOs in models of systems admitting large symmetry groups. The efficiency of these methods is based on development of the equivariant degree theory and its applications. This project will combine these new methods with a properly adapted version of Pyragas' control to achieve stabilization of UPOs in systems with complex symmetries. It is expected that these methods will be instrumental in navigating the system to various oscillation patterns (related to different spatio-temporal symmetries of UPOs) of practical interest. The application component of the project includes stabilization of UPOs in models of networks of electronic circuits, networks of hysteretic oscillators, and symmetrically coupled semiconductor lasers.