Generalized eigenvalue approach for dynamic mode decomposition

Traditional dynamic mode decomposition (DMD) methods inevitably involve matrix inversion, which often brings in numerical instability and spurious modes. In this paper, a new algorithm is derived to solve DMD as a general eigenvalue problem, which is then computed with a projection to a subspace with minimal errors in terms of least-squares (LS) or total-least-squares (TLS), leading to a more stable DMD algorithm, named DMD-LS or DMD-TLS, respectively. A new residual criterion, along with a typical energy criterion, is then proposed to select the most dynamically relevant DMD eigenvalues and modes. The accuracy and robustness of DMD-LS and DMD-TLS algorithms are demonstrated by application to the direct simulation data of the three-dimensional flow past a fixed long cylinder, which covers the entire evolution from the asymptotic periodic to the completely periodic stages of the flow. The connection between DMD modes and Floquet modes commonly used in stability studies was demonstrated through the DMD analysis of the asymptotic periodic data for the secondary instability of the flow.