HODMD: Higher Order Dynamic Mode Decomposition. This paper deals with an extension of dynamic mode decomposition (DMD), which is appropriate to treat general periodic and quasi-periodic dynamics, and transients decaying to periodic and quasi-periodic attractors, including cases (not accessible to standard DMD) that show limited spatial complexity but a very large number of involved frequencies. The extension, labeled as higher order dynamic mode decomposition, uses time-lagged snapshots and can be seen as superimposed DMD in a sliding window. The new method is illustrated and clarified using some toy model dynamics, the Stuart--Landau equation, and the Lorenz system. In addition, the new method is applied to (and its robustness is tested in) some permanent and transient dynamics resulting from the complex Ginzburg--Landau equation (a paradigm of pattern forming systems), for which standard DMD is seen to only uncover trivial dynamics, and the thermal convection in a rotating spherical shell subject to a radial gravity field.

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  1. Viguerie, Alex; Barros, Gabriel F.; Grave, Malú; Reali, Alessandro; Coutinho, Alvaro L. G. A.: Coupled and uncoupled dynamic mode decomposition in multi-compartmental systems with applications to epidemiological and additive manufacturing problems (2022)
  2. Gadalla, Mahmoud; Cianferra, Marta; Tezzele, Marco; Stabile, Giovanni; Mola, Andrea; Rozza, Gianluigi: On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis (2021)
  3. Lin, Yen Ting; Tian, Yifeng; Livescu, Daniel; Anghel, Marian: Data-driven learning for the Mori-Zwanzig formalism: a generalization of the Koopman learning framework (2021)
  4. Uy, Wayne Isaac Tan; Peherstorfer, Benjamin: Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectories (2021)
  5. Hijazi, Saddam; Stabile, Giovanni; Mola, Andrea; Rozza, Gianluigi: Data-driven POD-Galerkin reduced order model for turbulent flows (2020)
  6. Kamb, Mason; Kaiser, Eurika; Brunton, Steven L.; Kutz, J. Nathan: Time-delay observables for Koopman: theory and applications (2020)
  7. Le Clainche, Soledad; Vega, José M.: A review on reduced order modeling using DMD-based methods (2020)
  8. Pan, Shaowu; Duraisamy, Karthik: On the structure of time-delay embedding in linear models of non-linear dynamical systems (2020)
  9. Peherstorfer, Benjamin: Sampling low-dimensional Markovian dynamics for preasymptotically recovering reduced models from data with operator inference (2020)
  10. Champion, Kathleen P.; Brunton, Steven L.; Kutz, J. Nathan: Discovery of nonlinear multiscale systems: sampling strategies and embeddings (2019)
  11. Pascarella, G.; Fossati, M.; Barrenechea, G.: Adaptive reduced basis method for the reconstruction of unsteady vortex-dominated flows (2019)
  12. Le Clainche, Soledad; Vega, José M.: Spatio-temporal Koopman decomposition (2018)