论文标题
具有对称性的延迟微分方程的特征矩阵函数
Characteristic matrix functions for delay differential equations with symmetry
论文作者
论文摘要
特征矩阵函数捕获了矩阵值函数中有界线性运算符的光谱信息。在本文中，我们考虑一个具有一个离散时间延迟的延迟微分方程，并假设该方程相对于紧凑型对称组是均等的。在此假设下，延迟微分方程可以具有离散的波解，即具有离散的时空固定溶液。我们表明，如果离散波解的周期与时间延迟合理相关，那么我们可以使用特征矩阵函数来确定其稳定性。该证明依赖于浮力理论以及Kaashoek和Verduyn Lunel对紧凑型操作员类别的特征矩阵函数的结果。我们在延迟的周期轨道反馈稳定下讨论了结果的应用。
A characteristic matrix function captures the spectral information of a bounded linear operator in a matrix-valued function. In this article, we consider a delay differential equation with one discrete time delay and assume this equation is equivariant with respect to a compact symmetry group. Under this assumption, the delay differential equation can have discrete wave solutions, i.e. periodic solutions that have a discrete group of spatio-temporal symmetries. We show that if a discrete wave solution has a period that is rationally related to the time delay, then we can determine its stability using a characteristic matrix function. The proof relies on equivariant Floquet theory and results by Kaashoek and Verduyn Lunel on characteristic matrix functions for classes of compact operators. We discuss applications of our result in the context of delayed feedback stabilization of periodic orbits.
