Delay-coordinate maps, coherence, and approximate spectra of evolution operators
Delay-coordinate maps, coherence, and approximate spectra of evolution operators
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DOI：
10.1007/s40687-020-00239-y
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发表时间：
2021-03-01
影响因子：
1.2
通讯作者：
Giannakis, Dimitrios
中科院分区：
文献类型：
--
作者：
Giannakis, Dimitrios
利用核积分算子技术研究了保测遍历动力系统相干观测量的数据驱动辨识问题。提出了一种方法，即复值观测与近似周期性行为的构造从一对特征函数的积分算子建立从延迟坐标映射数据。结果表明，这些观测量是系统的Koopman演化算子的ε-近似本征函数，具有由延迟嵌入窗口的长度、演化时间和适当的谱隙参数控制的约束条件。特别地，当嵌入窗口增加时，可以使λ任意小，只要相应的本征值在积分算子的谱中保持足够孤立。它还表明，这种观测量的时间自相关函数是ε近似Koopman特征值，表现出一个定义良好的特征振荡频率（估计使用Koopman发电机）和一个缓慢衰减的调制包络。结果保持测量，遍历动力系统的任意谱特征，包括混合系统的连续谱和L-2中没有非常常数Koopman特征函数。数值例子揭示了一个相干的观测的Lorenz 63系统的自相关函数保持在0.5以上的模超过约10李雅普诺夫时间尺度。
The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair of eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are epsilon-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound epsilon controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, epsilon can be made arbitrarily small as the embedding window increases so long as the corresponding eigenvalues remain sufficiently isolated in the spectrum of the integral operator. It is also shown that the time-autocorrelation functions of such observables are epsilon-approximate Koopman eigenvalues, exhibiting a well-defined characteristic oscillatory frequency (estimated using the Koopman generator) and a slowly decaying modulating envelope. The results hold for measure-preserving, ergodic dynamical systems of arbitrary spectral character, including mixing systems with continuous spectrum and no non-constant Koopman eigenfunctions in L-2. Numerical examples reveal a coherent observable of the Lorenz 63 system whose autocorrelation function remains above 0.5 in modulus over approximately 10 Lyapunov timescales.