标题： 奇数定理的等变版本 显示英文标题

标题： Equivariant Versions of Odd Number Theorem

摘要： 我们考虑通过非侵入性延迟反馈控制，即Pyragas控制方法，来稳定自主系统的不稳定周期解的问题。 奇数定理对增益矩阵的选择施加了重要的限制，它指出了稳定的必要条件。 在本文中，奇数定理被扩展到等变系统。 我们假设未受控和受控系统都尊重一个对称群。 讨论了两种类型的结果。 首先，我们考虑旋转对称系统，其中控制稳定了在相空间中形成不变二维环面的相对周期解的整个轨道。 其次，我们考虑一种最近为具有有限对称群的系统提出的Pyragas控制方法的修改版本。 这种控制非侵入性地作用于轨道中的一个选定周期解，并旨在稳定该特定解。 针对上述两类系统，提出了奇数限制定理的不同变体。 结果通过一些先前在Pyragas控制文献中研究过的例子进行说明，包括两个对称耦合的Stewart-Landau振荡器系统和两个耦合激光器系统。 显示英文摘要

摘要： We consider the problem of stabilization of unstable periodic solutions to autonomous systems by the non-invasive delayed feedback control known as Pyragas control method. The Odd Number Theorem imposes an important restriction upon the choice of the gain matrix by stating a necessary condition for stabilization. In this paper, the Odd Number Theorem is extended to equivariant systems. We assume that both the uncontrolled and controlled systems respect a group of symmetries. Two types of results are discussed. First, we consider rotationally symmetric systems for which the control stabilizes the whole orbit of relative periodic solutions that form an invariant two-dimensional torus in the phase space. Second, we consider a modification of the Pyragas control method that has been recently proposed for systems with a finite symmetry group. This control acts non-invasively on one selected periodic solution from the orbit and targets to stabilize this particular solution. Variants of the Odd Number Limitation Theorem are proposed for both above types of systems. The results are illustrated with examples that have been previously studied in the literature on Pyragas control including a system of two symmetrically coupled Stewart-Landau oscillators and a system of two coupled lasers.