Koopman算子的谱、函数空间中的谱展开和状态空间几何
Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State Space Geometry
摘要：
我们研究了具有全局稳定吸引子的线性和非线性动力系统的谱算子理论性质。利用Kato分解，我们发展了一般线性自治动力系统的解析观测量的谱展开，并定义了相关的Koopman算子的广义本征函数的概念。我们解释稳定，不稳定和中心子空间的广义本征函数的零水平集。然后，我们利用共轭性质的Koopman特征函数和新的概念，开放的特征函数-定义在状态空间的子集-扩展这些结果的非线性动力系统的平衡。利用非线性系统的Koopman算子本征函数族的联合零水平集，给出了（全局）中心流形、中心稳定流形和中心不稳定流形的一个特征，定义了一类新的Hilbert空间，它刻画了耗散动力学的开和关吸引子性质，并引入了调制Fock空间的概念，我们发展了一类具有全局稳定极限环和极限环面的动力系统的谱展开式，其观测量在吸引子变量中是平方可积的，在吸引子变量中是解析的。我们讨论了这类具有（拟）-周期吸引子的非线性系统的稳定流形、不稳定流形和全局中心流形的定义。我们定义的概念的一般类的非线性系统的等稳。我们提供了一个简单的例子，保测系统是不是混沌，但有连续谱，并讨论了实验观察到的频谱在这样的系统。我们定义了一类数据集的相干主维的基础上相关联的Koopman算子的格型主谱。
英文摘要：
We examine spectral operator-theoretic properties of linear and nonlinear dynamical systems with globally stable attractors. Using the Kato Decomposition we develop a spectral expansion for general linear autonomous dynamical systems with analytic observables, and define the notion of generalized eigenfunctions of the associated Koopman operator. We interpret stable, unstable and center subspaces in terms of zero level sets of generalized eigenfunctions. We then utilize conjugacy properties of Koopman eigenfunctions and the new notion of open eigenfunctions - defined on subsets of state space - to extend these results to nonlinear dynamical systems with an equilibrium. We provide a characterization of (global) center manifolds, center-stable and center-unstable manifolds in terms of joint zero level sets of families of Koopman operator eigenfunctions associated with the nonlinear system.After defining a new class of Hilbert spaces, that capture the on and off attractor properties of dissipative dynamics, and introduce the concept of Modulated Fock Spaces}, we develop spectral expansions for a class of dynamical systems possessing globally stable limit cycles and limit tori, with observables that are square-integrable in on-attractor variables and analytic in off-attractor variables. We discuss definitions of stable, unstable and global center manifolds in such nonlinear systems with (quasi)-periodic attractors in terms of zero level sets of Koopman operator eigenfunctions. We define the notion of isostables for a general class of nonlinear systems. We provide a simple example of a measure-preserving system that is not chaotic but has continuous spectrum, and discuss experimental observations of spectrum on such systems. We define the coherent principal dimension for a class of datasets based on the lattice-type principal spectrum of the associated Koopman operator.
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