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The observation and study of nonlinear dynamical systems has been gaining popularity over years in different fields. The intrinsic complexity of their dynamics defies many existing tools based on individual orbits, while the Koopman operator governs evolution of functions defined in phase space and is thus focused on ensembles of orbits, which provides an alternative approach to investigate global features of system dynamics prescribed by spectral properties of the operator. However, it is difficult to identify and represent the most relevant eigenfunctions in practice. Here, combined with the Koopman analysis, a neural network is designed to achieve the reconstruction and evolution of complex dynamical systems. By invoking the error minimization, a fundamental set of Koopman eigenfunctions are derived, which may reproduce the input dynamics through a nonlinear transformation provided by the neural network. The corresponding eigenvalues are also directly extracted by the specific evolutionary structure built in. 相似文献
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Anderson localization is a famous wave phenomenon that describes the absence of diffusion of waves in a disordered medium. Here we generalize the landscape theory of Anderson localization to general elliptic operators and complex boundary conditions using a probabilistic approach, and further investigate some mathematical aspects of Anderson localization that are rarely discussed before. First, we observe that under the Neumann boundary condition, the low energy quantum states are localized on the boundary of the domain with high probability. We provide a detailed explanation of this phenomenon using the concept of extended subregions and obtain an analytical expression of this probability in the one-dimensional case. Second, we find that the quantum states may be localized in multiple different subregions with high probability in the one-dimensional case and we derive an explicit expression of this probability for various boundary conditions. Finally, we examine a bifurcation phenomenon of the localization subregion as the strength of disorder varies. The critical threshold of bifurcation is analytically computed based on a toy model and the dependence of the critical threshold on model parameters is analyzed. 相似文献
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从量子力学诞生日起,它的经典对应(或类比)一直是物理学家关心的话题.本文以介观电路量子化的框架中,带有互感的两个介观电容-电感(LC)电路为例,首次讨论了量子纠缠的经典类比(或对应)问题.先用有序算符内的积分理论证明其互感是产生量子纠缠的源头;再推导出求解特征频率的公式,就发现它与一个经典系统的小振动频率的表达式有相似之处,该经典系统组成如下:两个墙壁各连一个相同的弹簧,两个弹簧之间接着一个滑动小车可以在光滑的桌面上运动,小车挂有一根单摆.用分析力学求此系统的小振动频率,发现与上述介观电路的特征频率形式类似,单摆的摆动会造成小车来回振动,摆、小车和弹簧的互相牵制效应反映了小车和摆的"纠缠". 相似文献
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A field of endomorphisms R is called a Nijenhuis operator if its Nijenhuis torsion vanishes. In this work we study a specific kind of singular points of R called points of scalar type. We show that the tangent space at such points possesses a natural structure of a left-symmetric algebra (also known as pre-Lie or Vinberg-Kozul algebras). Following Weinstein's approach to linearization of Poisson structures, we state the linearisation problem for Nijenhuis operators and give an answer in terms of non-degenerate left-symmetric algebras. In particular, in dimension 2, we give classification of non-degenerate left-symmetric algebras for the smooth category and, with some small gaps, for the analytic one. These two cases, analytic and smooth, differ. We also obtain a complete classification of two-dimensional real left-symmetric algebras, which may be an interesting result on its own. 相似文献
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In this paper, we consider the non-autonomous semilinear impulsive differential equations with state-dependent delay. The approximate controllability results of the first-order systems are obtained in a separable reflexive Banach space, which has a uniformly convex dual. In order to establish sufficient conditions of the approximate controllability of such a system, we have used the theory of linear evolution systems, properties of the resolvent operator and Schauder’s fixed point theorem. Finally, we provide two concrete examples to validate our results. 相似文献
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