On the structure of time-delay embedding in linear models of non-linear dynamical systems
On the structure of time-delay embedding in linear models of non-linear dynamical systems
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DOI：
10.1063/5.0010886
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发表时间：
2020-07-01
期刊：
影响因子：
2.9
通讯作者：
Duraisamy, Karthik
中科院分区：
文献类型：
--
作者：
Pan, Shaowu;Duraisamy, Karthik
这项工作解决了有关的结构和条件的非线性动力学的吸引子的线性时滞模型的基本问题。虽然这种方法已经在渐近意义上得到了很好的研究（例如，对于无限数量的延迟），非渐近设置不是很好理解。首先，我们证明了完美的信号恢复所需的最小时间延迟完全由标量系统的傅立叶谱中的稀疏性决定。对于向量的情况下，我们提供了一个秩检验和几何解释的必要条件和充分条件存在一个精确的线性时滞模型。此外，我们证明了由傅立叶谱引起的线性系统的输出可控性指标作为所需的最小时滞数的紧上界。还提供了谱域中精确线性模型的显式表达式。从数值的角度来看，采样率和数值条件的时间延迟的数量的影响进行检查。上界的条件数推导出，与暗示，条件可以改善额外的时间延迟和/或降低采样率。此外，它明确地表明，基本的动态可以准确地恢复仅使用部分周期的吸引子。我们的分析首先在简单的周期和准周期系统中进行了验证，并对噪声的敏感性进行了研究。最后，讨论了大规模混沌系统中时滞的选择问题和实用策略，并以三维湍流Rayleigh-Benard对流为例进行了演示。由AIP Publishing授权出版。
This work addresses fundamental issues related to the structure and conditioning of linear time-delayed models of non-linear dynamics on an attractor. While this approach has been well-studied in the asymptotic sense (e.g., for an infinite number of delays), the non-asymptotic setting is not well-understood. First, we show that the minimal time-delays required for perfect signal recovery are solely determined by the sparsity in the Fourier spectrum for scalar systems. For the vector case, we provide a rank test and a geometric interpretation for the necessary and sufficient conditions for the existence of an accurate linear time delayed model. Furthermore, we prove that the output controllability index of a linear system induced by the Fourier spectrum serves as a tight upper bound on the minimal number of time delays required. An explicit expression for the exact linear model in the spectral domain is also provided. From a numerical perspective, the effect of the sampling rate and the number of time delays on numerical conditioning is examined. An upper bound on the condition number is derived, with the implication that conditioning can be improved with additional time delays and/or decreasing sampling rates. Moreover, it is explicitly shown that the underlying dynamics can be accurately recovered using only a partial period of the attractor. Our analysis is first validated in simple periodic and quasiperiodic systems, and sensitivity to noise is also investigated. Finally, issues and practical strategies of choosing time delays in large-scale chaotic systems are discussed and demonstrated on 3D turbulent Rayleigh-Benard convection. Published under license by AIP Publishing.