Applied Koopmanism

Chaos
Marko BudisićIgor Mezić

Abstract

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. The first goal of this paper is to make it clear how methods that appeared in different papers and contexts all relate to each other through spectral properties of the Koopman operator. The second goal is to present these methods in a concise manner in an effort to make the framework accessible to researchers who would like to apply them, but also, expand and improve them. Finally, we aim to provide a road map through the literature where each of the topics was described in detail. We describe three main concepts: Koopman mode analysis, Koop...Continue Reading

References

May 19, 2005·Proceedings of the National Academy of Sciences of the United States of America·R R CoifmanS W Zucker
May 1, 1931·Proceedings of the National Academy of Sciences of the United States of America·B O Koopman
Nov 6, 2008·Medical Image Computing and Computer-assisted Intervention : MICCAI·Yonggang ShiArthur W Toga

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Citations

Jan 3, 2015·Chaos·David J Chappell, Gregor Tanner
Nov 4, 2015·Journal of Neuroscience Methods·Bingni W BruntonJ Nathan Kutz
Sep 4, 2015·Chaos·Matthew O WilliamsClarence W Rowley
Oct 3, 2015·Chaos·J D Meiss
Oct 11, 2017·Proceedings. Mathematical, Physical, and Engineering Sciences·Ayoub GouasmiKarthik Duraisamy
Jul 25, 2019·Physical Review. E·Gerard McCaulDenys I Bondar
Oct 16, 2019·Proceedings. Mathematical, Physical, and Engineering Sciences·Denys I BondarCesare Tronci
Oct 28, 2019·Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences·Yoshiki Kuramoto, Hiroya Nakao
Oct 3, 2019·Chaos·Anastasiya SalovaRaissa M D'Souza
Feb 12, 2017·Physical Review Letters·Andreas OttoGünter Radons
Jan 20, 2018·Physical Review. E·Naoya TakeishiTakehisa Yairi
Nov 25, 2018·Nature Communications·Bethany LuschSteven L Brunton
Mar 7, 2019·Proceedings. Mathematical, Physical, and Engineering Sciences·E KaiserS L Brunton
May 3, 2020·Chaos·Felix DietrichIoannis G Kevrekidis
Mar 18, 2016·Physical Review. E·Adam SvenkesonBruce J West
Oct 2, 2017·Chaos·Mickaël D Chekroun, Dmitri Kondrashov
Jul 22, 2020·Physical Review. E·William T Redman
Jul 31, 2020·Data Mining and Knowledge Discovery·Wei ZhangJr-Shin Li
Jun 1, 2017·Nature Communications·Steven L BruntonJ Nathan Kutz
Jul 16, 2017·Physical Review. E·David MüllerGünter Radons
Oct 24, 2019·Physical Review. E·Dimitrios Giannakis
May 17, 2017·Science Advances·Samuel H RudyJ Nathan Kutz
Mar 16, 2018·Physical Review Letters·David MüllerGünter Radons
Nov 21, 2020·Proceedings. Mathematical, Physical, and Engineering Sciences·Kadierdan KahemanSteven L Brunton
Jan 1, 2018·SIAM Journal on Applied Dynamical Systems·Joshua L ProctoryJ Nathan Kutzx

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