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

自然基本定律的对称性表现在自然现象中大量的对称性上。通过相互作用振荡器的网络对这种现象进行建模在从电子学和光通信到神经学和医学的各种科学和技术领域中是常见的。控制这种网络中自发产生的振荡模式的对称特征构成了一个重大挑战。这个项目的目标是开发新的方法，允许简单而有效的控制复杂的对称网络的振荡器的性能。这项工作的重点是非侵入性的控制方法，允许实现所需的行为与最小的强制系统，可以很容易地采用实际的实施（例如，在电子电路）。该项目的教育部分旨在让从本科到研究生水平的学生积极参与这项研究，研究具有特定对称群的振荡器的具体网络。不稳定系统的镇定是应用非线性科学中的重要问题之一。真实的生命系统的自然对称性导致了多个周期区域，使稳定问题复杂化。本研究的目的是发展的方法来稳定不稳定的周期性制度的对称系统的延迟反馈控制和线性比例控制相结合。该项目的重点将是两种类型的对称系统的记忆，即滞后和延迟系统。非侵入性的延迟反馈控制（由于Pyragas）的不稳定周期轨道（UPO）的稳定方法已成功地测试对不同类型的混沌系统模型。在过去的两年中，co-PI一直在开发计算机化的符号数值工具，并支持理论框架的本地化和分类的UPO的系统模型承认大的对称群。这些方法的有效性是基于等变度理论及其应用的发展。该项目将联合收割机这些新的方法与适当适应版本的金字塔控制，以实现稳定的不稳定的系统与复杂的对称性。预计这些方法将有助于导航系统的各种振荡模式（有关不同的时空对称性的UPO）的实际利益。该项目的应用部分包括稳定的UPO的网络模型的电子电路，滞后振荡器网络，对称耦合半导体激光器。

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.