发展算子的延迟坐标映射、相干性和近似谱
Delay-coordinate maps, coherence, and approximate spectra of evolution operators
摘要：
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本征函数的积分算子从延迟坐标映射的数据。结果表明，这些可观的$ε$ -近似的Koopman系统的演化算子的本征函数，与限制$ε$控制的延迟嵌入窗口的长度，演化时间，和适当的谱隙参数。特别地，$ ε$可以随着嵌入窗口的增加而任意小，只要相应的本征值在积分算子的谱中保持充分隔离。它还表明，这种观测量的时间自相关函数是$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）和缓慢衰减的调制包络。这些结果适用于任意谱特征的保测遍历动力系统，包括具有连续谱和$L^2$中没有非常数Koopman本征函数的混合系统.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。
英文摘要：
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 eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are $ε$-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound $ε$ controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, $ ε$ 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 $ε$-approximate Koopman eigenvalue, 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.
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