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

自然基本规律的对称性体现在自然现象中的大量对称性上。通过相互作用的振荡器网络对这种现象进行建模在从电子和光学通信到神经学和医学的各个科学和技术领域中都很常见。在这样的网络中控制自发产生的振荡图案的对称特征是一个巨大的挑战。这个项目的目标是开发新的方法，允许简单而有效地控制振荡器的复杂对称网络的性能。这项工作的重点是非侵入性的控制方法，这些方法允许在最小限度地强迫系统的情况下实现所需的行为，并且可以容易地用于实际实施(例如，在电子电路中)。这个项目的教育部分旨在让从本科生到研究生的学生积极参与这项研究，研究具有特定对称群的振子的具体网络。不稳定系统的镇定是应用非线性科学中最重要的问题之一。现实系统的自然对称性导致了多重周期机制，并使镇定问题复杂化。本研究的目的是发展一种结合延迟反馈控制和线性比例控制的方法来镇定对称系统的不稳定周期区域。该项目的重点将放在两类具有记忆的对称系统上，即滞后系统和时滞系统。非侵入性延迟反馈控制不稳定周期轨道(UPO)的方法已经在不同类型的混沌系统模型上被成功地测试。在过去的两年中，合作的PIS一直在开发计算机化的符号数值工具和支持的理论框架，用于在接纳大对称群的系统的模型中定位和分类UPO。这些方法的有效性是基于等变度理论及其应用的发展。这个项目将把这些新的方法与适当调整的版本的火焰气控制相结合，以实现复杂对称性系统中的UPO的稳定。预计这些方法将有助于将系统导航到实际感兴趣的各种振荡模式(与UPO的不同时空对称性有关)。该项目的应用部分包括在电子电路网络模型、滞后振荡器网络模型和对称耦合半导体激光器模型中稳定UPO。

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.