论文标题
非自主线性动力学系统的角度值：第二部分 - 还原理论和算法
Angular Values of Nonautonomous Linear Dynamical Systems: Part II -Reduction Theory and Algorithm
论文作者
论文摘要
这项工作着重于非自主动力学系统的角值，这些动力系统已在先前出版物中为一般随机和（非）自主动力系统引入[W.-J. Beyn，G。Froyland和T.Hüls，Siam J. Appl。 dyn。 Syst。，21（2022），第1245---1286页。尺寸$ s $的角值测量了通过离散时间线性系统的动力学的相位空间体验的最大平均旋转。我们的主要结果将角值的概念与众所周知的二分法（或Sacker-sell）频谱及其相关的光谱束有关。特别是，我们证明了一个还原定理，该定理表明，而不是在所有子空间上最大化，而是在光谱纤维中具有基础的所谓痕量空间上最大化。还原导致一种用于计算尺寸的角度值的算法。我们将算法应用于多达4个尺寸系统的算法，并证明其效率以检测最快的旋转子空间，即使在正向动力学下它不主导。
This work focuses on angular values of nonautonomous dynamical systems which have been introduced for general random and (non)autonomous dynamical systems in a previous publication [W.-J. Beyn, G. Froyland, and T. Hüls, SIAM J. Appl. Dyn. Syst., 21 (2022), pp. 1245--1286]. The angular value of dimension $s$ measures the maximal average rotation which an $s$-dimensional subspace of the phase space experiences through the dynamics of a discrete-time linear system. Our main results relate the notion of angular value to the well-known dichotomy (or Sacker--Sell) spectrum and its associated spectral bundles. In particular, we prove a reduction theorem which shows that instead of maximizing over all subspaces, it suffices to maximize over so-called trace spaces which have their basis in the spectral fibers. The reduction leads to an algorithm for computing angular values of dimensions one and two. We apply the algorithm to several systems of dimension up to 4 and demonstrate its efficiency to detect the fastest rotating subspace even if it is not dominant under the forward dynamics.
