From dynamical systems with time-varying delay to circle maps and Koopman operators.
From dynamical systems with time-varying delay to circle maps and Koopman operators.
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从具有时变延迟的动力系统到圆图和库普曼算子
DOI：
10.1103/physreve.95.062214
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发表时间：
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通讯作者：
G. Radons
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文献类型：
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作者：
D. Müller;A. Otto;G. Radons
本文研究了时变时滞的延迟访问对时滞系统动力学的影响。我们发现，有两个普遍性类的延迟，导致根本的差异，如李雅普诺夫谱的动力学量。因此，我们引入了一个算子理论框架，其中的解决方案运营商的延迟系统被分解为Koopman运营商描述的延迟访问和运营商类似的解决方案运营商已知的系统具有常数延迟。Koopman算子对应于一个迭代映射，称为访问映射，它由延迟方程的延迟自变量的迭代定义。这个一维迭代映射的动力学性质决定了由时滞微分方程支配的无限维状态动力学的普适类。通过这种方式，我们将时滞系统理论与圆映射理论和Koopman算子框架联系起来。本文推广了我们以前的工作[A. Otto，D. Müller和G. Radons，Phys. Rev. Lett. 118，044104（2017）PRLTAO 0031 -900710.1103/PhysRevLett.118.044104]，详细阐述了数学细节，并提出了关于李雅普诺夫向量的进一步结果。
In this paper, we investigate the influence of the retarded access by a time-varying delay on the dynamics of delay systems. We show that there are two universality classes of delays, which lead to fundamental differences in dynamical quantities such as the Lyapunov spectrum. Therefore, we introduce an operator theoretic framework, where the solution operator of the delay system is decomposed into the Koopman operator describing the delay access and an operator similar to the solution operator known from systems with constant delay. The Koopman operator corresponds to an iterated map, called access map, which is defined by the iteration of the delayed argument of the delay equation. The dynamics of this one-dimensional iterated map determines the universality classes of the infinite-dimensional state dynamics governed by the delay differential equation. In this way, we connect the theory of time-delay systems with the theory of circle maps and the framework of the Koopman operator. In this paper, we extend our previous work [A. Otto, D. Müller, and G. Radons, Phys. Rev. Lett. 118, 044104 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.044104] by elaborating the mathematical details and presenting further results also on the Lyapunov vectors.