Singular Values of Hankel Matrix: A Practical Method to Identify the Koopman Eigenfunctions.

To infer eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator, we study the leading singular values of the Hankel matrix associated with a given observable of a dynamical system. We prove that for a discrete-time measure preserving ergodic dynamic system and a given square-integrable observable $f$, these leading singular values of the Hankel matrix have one-to-one correspondence with the energy of $f$ represented by the Koopman eigenfunctions. The proof is associated to the Birkhoff ergodic theorem, 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. 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 the leading temporal empirical orthogonal functions are indeed approximations of the Koopman eigenfunctions. A theorem-based practical methodology is then proposed to identify the Koopman eigenfunctions from a given time series. It builds on the fact that the convergence of the renormalized leading singular values of the Hankel matrix is a necessary and sufficient condition for the existence of Koopman eigenfunctions. Numerical illustrating results on simple low dimensional systems and real interpolated ocean sea-surface height data are presented and discussed.