TY - JOUR T1 - Eigenvalues of autocovariance matrix: A practical method to identify the Koopman eigenfrequencies A1 - Zhen,Yicun A1 - Chapron,Bertrand A1 - Mémin,Etienne A1 - Peng,Lin AD - Institut Franais de Recherche pour l'Exploitation de la Mer, 29280 Plouzané, France AD - INRIA/IRMAR, Campus universitaire de Beaulieu, Rennes, 35042 Cedex, France AD - Ocean University of China, 266100 Qingdao, China UR - https://archimer.ifremer.fr/doc/00764/87649/ DO - 10.1103/PhysRevE.105.034205 N2 - To infer eigenvalues of the infinite-dimensional Koopman operator, we study the leading eigenvalues of the autocovariance matrix associated with a given observable of a dynamical system. For any observable f for which all the time-delayed autocovariance exist, we construct a Hilbert space Hf and a Koopman-like operator K that acts on Hf. We prove that the leading eigenvalues of the autocovariance matrix has one-to-one correspondence with the energy of f that is represented by the eigenvectors of K. The proof is associated to several representation theorems of isometric operators on a Hilbert space, and the weak-mixing property of the observables represented by the continuous spectrum. We also provide an alternative proof of the weakly mixing property. When f is an observable of an ergodic dynamical system which has a finite invariant measure μ, Hf coincides with closure in L2(X,dμ) of Krylov subspace generated by f, and K coincides with the classical Koopman operator. The main theorem sheds light to the theoretical foundation of several semi-empirical methods, including singular spectrum analysis (SSA), data-adaptive harmonic analysis (DAHD), Hankel DMD, and Hankel alternative view of Koopman analysis (HAVOK). It shows that, when the system is ergodic and has finite invariant measure, the leading temporal empirical orthogonal functions indeed correspond to the Koopman eigenfrequencies. A theorem-based practical methodology is then proposed to identify the eigenfrequencies of K from a given time series. It builds on the fact that the convergence of the renormalized eigenvalues of the Gram matrix is a necessary and sufficient condition for the existence of K−eigenfrequencies. Numerical illustrating results on simple low dimensional systems and real interpolated ocean sea-surface height data are presented and discussed. Y1 - 2022/03 PB - American Physical Society (APS) JF - Physical Review E SN - 2470-0045 VL - 105 IS - 3 ID - 87649 ER -