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首页> 外文期刊>SIAM journal on applied dynamical systems >Koopman Operator Family Spectrum for Nonautonomous Systems
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Koopman Operator Family Spectrum for Nonautonomous Systems

机译：Koopman操作员非自来组织系统的家庭谱

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摘要

For any nonautonomous dynamical system, the family of Koopman operators, as well as related Koopman eigenvalues and eigenfunctions, is parameterized by a time pair. Therefore, a logical approach in the data-driven algorithms for the nonautonomous Koopman mode decomposition is the application of a dynamic mode decomposition (DMD) method on the moving stencils of snapshots in order to capture the time dependency. In this paper, we investigate the issues that arise in such an approach. These issues do not appear if we use the moving stencil approach as the model fitting method; they appear as significant errors in the computed nonautonomous Koopman operator eigenvalues. The first issue manifests itself in the hybrid dynamical systems when the moving stencil passes over a nonautonomous switching point. We show that such stencils can be detected through the Krylov subspace projection error and propose an algorithm that computes correct eigenvalues by avoiding such stencils. The second issue appears in the continuous-in-time nonautonomous systems. Even if we apply techniques of finding good observables that solve all issues in the autonomous case, the nonautonomous Koopman eigenvalues will still be computed with a significant error. In the presented theorems, we reveal the nature of this error and propose a second algorithm that is based on the reduction of the stencil size. The application of the two new data-driven algorithms on various nonautonomous systems shows complete recovery from the errors otherwise present in computation of the nonautonomous Koopman operator eigenvalues.
机译：对于任何非自治动态系统，Koopman运算符的家族以及相关的Koopan特征值和特征障碍都是通过时间对参数化的。因此，用于非自治的Koopman模式分解的数据驱动算法中的逻辑方法是在快照的移动模板上应用动态模式分解（DMD）方法，以捕获时间依赖性。在本文中，我们调查了这种方法中出现的问题。如果我们使用移动的模板方法作为模型拟合方法，则不会出现这些问题;它们在计算的非自治的Koopman操作员特征值中表现出重大错误。当移动的模板通过非编程切换点时，第一问题在混合动态系统中表现出混合动态系统。我们表明，可以通过Krylov子空间投影误差检测这种模板，并提出通过避免这种模板来计算正确的特征值的算法。第二个问题出现在连续立即非自治系统中。即使我们应用了解解决自主案例中的所有问题的良好观察的技术，仍将使用显着的错误计算非自治的Koopan特征值。在呈现的定理中，我们揭示了这个错误的性质，并提出了一种基于模版尺寸的减少的第二算法。两个新的数据驱动算法在各种非自治系统上的应用显示了在非自治的Koopman操作员特征值计算中出现的错误的完全恢复。

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