# Dynamical systems via Koopman operators

## Composition operators, Dynamic Extended Mode decompositions…

NB: Koopman here is B.O. Koopman not S.J. Koopman, who also works in dynamical systems.

I do not know how this works, but maybe this fragment of abstract will do for now :

A majority of methods from dynamical system analysis, especially those in applied settings, rely on Poincaré’s geometric picture that focuses on “dynamics of states.” While this picture has fueled our field for a century, it has shown difficulties in handling high-dimensional, ill-described, and uncertain systems, which are more and more common in engineered systems design and analysis of “big data” measurements. This overview article presents an alternative framework for dynamical systems, based on the “dynamics of observables” picture. The central object is the Koopman operator: an infinite-dimensional, linear operator that is nonetheless capable of capturing the full nonlinear dynamics.

There is a brief literature review in .

The Chaos book folks describe it so:

the Koopman operator action on a state space function $$a(x)$$ is to replace it by its downstream value time $$t$$ later, $$a(x) \rightarrow a(x(t)),$$ evaluated at the trajectory point $$x(t)$$

$\left[\mathcal{K}^{t} a\right](x)=a\left(f^{t}(x)\right)=\int_{\mathcal{M}} d y \mathcal{K}^{t}(x, y) a(y),$ $$\mathcal{K}^{t}(x, y)=\delta\left(y-f^{t}(x)\right)$$

## References

Arbabi, Hassan. 2020. “Introduction to Koopman Operator Theory of Dynamical Systems,” 32. https://www.mit.edu/~arbabi/research/KoopmanIntro.pdf.
Brunton, Steven L. 2019. “Notes on Koopman Operator Theory.” https://fluids.ac.uk/files/meetings/KoopmanNotes.1575558616.pdf.
Brunton, Steven L., and Jose Nathan Kutz. 2019. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge: Cambridge University Press. https://databookuw.com.
Brunton, Steven L., Joshua L. Proctor, and J. Nathan Kutz. 2016. “Discovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems.” Proceedings of the National Academy of Sciences 113 (15): 3932–37. https://doi.org/10.1073/pnas.1517384113.
Budišić, Marko, Ryan Mohr, and Igor Mezić. 2012. “Applied Koopmanism.” Chaos: An Interdisciplinary Journal of Nonlinear Science 22 (4): 047510. https://doi.org/10.1063/1.4772195.
Cvitanović, P., R. Artuso, R. Mainieri, G. Tanner, and G. Vattay. 2016. “Koopman Modes.” In Chaos: Classical and Quantum. Copenhagen: Niels Bohr Inst. http://ChaosBook.org/.
Ishikawa, Isao, Keisuke Fujii, Masahiro Ikeda, Yuka Hashimoto, and Yoshinobu Kawahara. 2018. “Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators.” arXiv:1805.12324 [cs, Math, Stat], October. http://arxiv.org/abs/1805.12324.
Klus, Stefan, Péter Koltai, and Christof Schütte. 2016. “On the Numerical Approximation of the Perron-Frobenius and Koopman Operator.” Journal of Computational Dynamics 3 (1): 51. https://doi.org/10.3934/jcd.2016003.
Klus, Stefan, Feliks Nüske, Sebastian Peitz, Jan-Hendrik Niemann, Cecilia Clementi, and Christof Schütte. 2020. “Data-Driven Approximation of the Koopman Generator: Model Reduction, System Identification, and Control.” Physica D: Nonlinear Phenomena 406 (May): 132416. https://doi.org/10.1016/j.physd.2020.132416.
Klus, Stefan, Ingmar Schuster, and Krikamol Muandet. 2020. “Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces.” Journal of Nonlinear Science 30 (1): 283–315. https://doi.org/10.1007/s00332-019-09574-z.
Koopman, B. O. 1931. “Hamiltonian Systems and Transformation in Hilbert Space.” Proceedings of the National Academy of Sciences 17 (5): 315–18. https://doi.org/10.1073/pnas.17.5.315.
Kutz, J. Nathan, Steven L. Brunton, Bingni W. Brunton, and Joshua L. Proctor. 2016. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems. Philadelphia, PA: Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611974508.
Li, Qianxiao, Felix Dietrich, Erik M. Bollt, and Ioannis G. Kevrekidis. 2017. “Extended Dynamic Mode Decomposition with Dictionary Learning: A Data-Driven Adaptive Spectral Decomposition of the Koopman Operator.” Chaos: An Interdisciplinary Journal of Nonlinear Science 27 (10): 103111. https://doi.org/10.1063/1.4993854.
Lin, Yen Ting, Yifeng Tian, Daniel Livescu, and Marian Anghel. 2021. “Data-Driven Learning for the Mori–Zwanzig Formalism: A Generalization of the Koopman Learning Framework.” arXiv:2101.05873 [cond-Mat], January. http://arxiv.org/abs/2101.05873.
Lusch, Bethany, J. Nathan Kutz, and Steven L. Brunton. 2018. “Deep Learning for Universal Linear Embeddings of Nonlinear Dynamics.” Nature Communications 9 (1): 4950. https://doi.org/10.1038/s41467-018-07210-0.
Mauroy, Alexandre, and Jorge Goncalves. 2020. “Koopman-Based Lifting Techniques for Nonlinear Systems Identification.” IEEE Transactions on Automatic Control 65 (6): 2550–65. https://doi.org/10.1109/TAC.2019.2941433.
Morrill, James, Patrick Kidger, Cristopher Salvi, James Foster, and Terry Lyons. 2020. “Neural CDEs for Long Time Series via the Log-ODE Method.” In, 5.
Schmid, Peter J. 2010. “Dynamic Mode Decomposition of Numerical and Experimental Data.” Journal of Fluid Mechanics 656 (August): 5–28. https://doi.org/10.1017/S0022112010001217.
Schwantes, Christian R., and Vijay S. Pande. 2015. “Modeling Molecular Kinetics with tICA and the Kernel Trick.” Journal of Chemical Theory and Computation 11 (2): 600–608. https://doi.org/10.1021/ct5007357.
Tu, Jonathan H., Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, and J. Nathan Kutz. 2014. “On Dynamic Mode Decomposition: Theory and Applications.” Journal of Computational Dynamics 1 (2): 391. https://doi.org/10.3934/jcd.2014.1.391.
Williams, Matthew O., Ioannis G. Kevrekidis, and Clarence W. Rowley. 2015. “A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition.” Journal of Nonlinear Science 25 (6): 1307–46. https://doi.org/10.1007/s00332-015-9258-5.
Williams, Matthew O., Clarence W. Rowley, and Ioannis G. Kevrekidis. 2015. “A Kernel-Based Method for Data-Driven Koopman Spectral Analysis.” Journal of Computational Dynamics 2 (2): 247. https://doi.org/10.3934/jcd.2015005.

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