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Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems [article]

Yoshihiko Susuki, Alexandre Mauroy, Igor Mezic
2021 arXiv   pre-print
This shows that the Koopman resolvent provides the Laplace-domain representation of such nonlinear autonomous dynamics.  ...  The motivation of our research is to establish a Laplace-domain theory that provides principles and methodology to analyze and synthesize systems with nonlinear dynamics.  ...  The Laplace domain is a classical approach to analysis and synthesis of linear autonomous systems: see, Table 1 : Laplace-domain representation of nonlinear autonomous dynamical systems studied in this  ... 
arXiv:2009.11544v2 fatcat:cc5nljq46zds7lwhuu5s6ppmvy

Modern Koopman Theory for Dynamical Systems [article]

Steven L. Brunton, Marko Budišić, Eurika Kaiser, J. Nathan Kutz
2021 arXiv   pre-print
Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinite-dimensional linear operator acting on the space of  ...  The success of Koopman analysis is due primarily to three key factors: 1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems, 2) the approach is formulated  ...  We also thank Shervin Bagheri, Bing Brunton, Bethany Lusch, Ryan Mohr, Frank Noe, Josh Proctor, Clancy Rowley, and Peter Schmid for many fruitful discussions on DMD, Koopman theory, and control.  ... 
arXiv:2102.12086v2 fatcat:2oylyx25dbctvkjfnirfcgjuqu

Koopman Operator Dynamical Models: Learning, Analysis and Control [article]

Petar Bevanda and Stefan Sosnowski and Sandra Hirche
2021 arXiv   pre-print
The Koopman operator allows for handling nonlinear systems through a (globally) linear representation.  ...  Also, Koopman operator theory has long-standing connections to known system-theoretic and dynamical system notions that are not universally recognized.  ...  For a connection of the operator's spectrum and it's resolvent generalizing Laplace-domain theory to nonlinear dynamics, we point to Susuki et al. [30] . Property 2.4 (Koopman eigenfunction group).  ... 
arXiv:2102.02522v1 fatcat:wtjh3w6xmvbkvg4lz2cncmehyy

Koopman Operator, Geometry, and Learning of Dynamical Systems

Igor Mezić
2021 Notices of the American Mathematical Society  
This paper contains only a limited number of references as journal rules restrict these to 20. The fully referenced version is posted on arXiv, as arXiv:2010.05377.  ...  of the Koopman operator associated with a dynamical system has a continuous part.  ...  The goal was to connect system inputs to system outputs via analysis of a structured model connecting these.  ... 
doi:10.1090/noti2306 fatcat:mjrjcdberff6fot2ckioas5mni

Time-Delay Observables for Koopman: Theory and Applications [article]

Mason Kamb, Eurika Kaiser, Steven L. Brunton, J. Nathan Kutz
2020 arXiv   pre-print
Nonlinear dynamical systems are ubiquitous in science and engineering, yet analysis and prediction of these systems remains a challenge.  ...  In this work we consider time-delay observables to represent nonlinear dynamics in the Koopman operator framework.  ...  JNK acknowledges support from the Air Force Office of Scientific Research (AFOSR) grant FA9550-17-1-0329.  ... 
arXiv:1810.01479v2 fatcat:atbg2kvl5fa33pt26hzcp2obxa

Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism [article]

Kevin K. Lin, Fei Lu
2020 arXiv   pre-print
We give a heuristic derivation of a NARMAX (Nonlinear Auto-Regressive Moving Average with eXogenous input) model from an underlying dynamical model.  ...  Here, we study data-driven model reduction from a dynamical systems perspective.  ...  We thank the Mathematics Group at Lawrence Berkeley National Lab for its support of this work; to Alexandre Chorin for many useful comments on the manuscript; and to Xiantao Li for encouraging us to study  ... 
arXiv:1908.07725v5 fatcat:tzzvfzecbrg6jop6fjcrsy7guq

Extraction and Prediction of Coherent Patterns in Incompressible Flows through Space-Time Koopman Analysis [article]

Dimitrios Giannakis, Suddhasattwa Das
2017 arXiv   pre-print
space-time coherent patterns under a skew-product dynamical system.  ...  We develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time-dependent fluid flows driven by ergodic dynamical systems.  ...  Suddhasattwa Das is supported as a postdoctoral research fellow from the first grant. The authors wish to thank Pierre Germain and Igor Mezić for stimulating conversations.  ... 
arXiv:1706.06450v1 fatcat:lmsbaeqb7fhbbnaamonxpbcohq

Pseudo generators for under-resolved molecular dynamics

A. Bittracher, C. Hartmann, O. Junge, P. Koltai
2015 The European Physical Journal Special Topics  
Many features of a molecule which are of physical interest (e.g. molecular conformations, reaction rates) are described in terms of its dynamics in configuration space.  ...  govern the dynamics of the position coordinate (without any assumptions on the damping).  ...  Trajectory-and ensemble-based views We consider a dynamical system described by d " 3n positional degrees of freedom that represent a system of n particles.  ... 
doi:10.1140/epjst/e2015-02422-y fatcat:j4z2nqzvhvhyrn3tfyuw5575oi

Modal Analysis of Fluid Flows: An Overview [article]

Kunihiko Taira, Steven L. Brunton, Scott T. M. Dawson, Clarence W. Rowley, Tim Colonius, Beverley J. McKeon, Oliver T. Schmidt, Stanislav Gordeyev, Vassilios Theofilis, Lawrence S. Ukeiley
2017 arXiv   pre-print
analysis, global linear stability analysis, and resolvent analysis.  ...  This step typically starts with a modal decomposition of an experimental or numerical dataset of the flow field, or of an operator relevant to the system.  ...  Acknowledgments This paper was one of the major outcomes from the AIAA Discussion Group (Fluid Dynamics Technical Committee) entitled "Modal Decomposition of Aerodynamics Flows" organized by KT and Dr.  ... 
arXiv:1702.01453v2 fatcat:3s6lovtg3bdzbnoodqy24t4iv4

Linear iterative method for closed-loop control of quasiperiodic flows

Colin Leclercq, Fabrice Demourant, Charles Poussot-Vassal, Denis Sipp
2019 Journal of Fluid Mechanics  
This work proposes a feedback-loop strategy to suppress intrinsic oscillations of resonating flows in the fully nonlinear regime.  ...  The frequency response of the flow is obtained from the resolvent operator about the mean flow, extending the framework initially introduced by McKeon & Sharma (J.  ...  The authors are also grateful to Matthew Juniper for sharing his Matlab toolbox for the analysis of nonlinear dynamics (Juniper & Sujith 2018) .  ... 
doi:10.1017/jfm.2019.112 fatcat:zfw2dtx7vnh3viibabe3r6ccta

Data-driven approximations of dynamical systems operators for control [article]

Eurika Kaiser, J. Nathan Kutz, Steven L. Brunton
2019 arXiv   pre-print
The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics.  ...  Koopman operator.  ...  JNK acknowledges support from the Air Force Office of Scientific Research (FA9550-19-1-0011).  ... 
arXiv:1902.10239v1 fatcat:23bw5qdalbenddnelslxtmes7m

Geometry of the ergodic quotient reveals coherent structures in flows

Marko Budišić, Igor Mezić
2012 Physica D : Non-linear phenomena  
Dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach.  ...  By segmenting the ergodic quotient based on the diffusion modes, we extract coherent features in the state space of the dynamical system.  ...  Let For a non-autonomous dynamical system defined by an ODEẋ = A(x, t) , where x ∈ M, we can define the extended, autonomous system througḣ x = A(x, τ ), τ = c, where (x, τ ) ∈ M e and c is a time-rescaling  ... 
doi:10.1016/j.physd.2012.04.006 fatcat:coz7gf3cqvctdm5xnsmudfn33m

The Occupation Kernel Method for Nonlinear System Identification [article]

Joel A. Rosenfeld, Benjamin Russo, Rushikesh Kamalapurkar, Taylor T. Johnson
2021 arXiv   pre-print
This framework allows for trajectories of a nonlinear dynamical system to be treated as a fundamental unit of data for nonlinear system identification routine.  ...  The approach to nonlinear system identification is demonstrated to identify parameters of a dynamical system accurately, while also exhibiting a certain robustness to noise.  ...  While many classical tools are available for system identification for linear dynamics using the impulse response and Fourier and Laplace transforms of the dynamical system, the identification of nonlinear  ... 
arXiv:1909.11792v3 fatcat:tnk7sz2izfeo5iusua7xj2smtq

Machine Learning for Prediction with Missing Dynamics [article]

John Harlim and Shixiao W. Jiang and Senwei Liang and Haizhao Yang
2020 arXiv   pre-print
This article presents a general framework for recovering missing dynamical systems using available data and machine learning techniques.  ...  components in the resolved dynamics.  ...  For dynamical systems driven by stochastic noises, one can rewrite the full dynamics as an autonomous dynamical system by augmenting (x n , y n ) with the entire history of the noises.  ... 
arXiv:1910.05861v3 fatcat:ancqfy3dgvaqniha5hecdgdrnu

Mean Subtraction and Mode Selection in Dynamic Mode Decomposition [article]

Gowtham S Seenivasaharagavan, Milan Korda, Hassan Arbabi, Igor Mezić
2021 arXiv   pre-print
Koopman mode analysis has provided a framework for analysis of nonlinear phenomena across a plethora of fields.  ...  We prove that this equivalence is impossible when the order of the DMD-based representation of the dynamics exceeds the dimension of the system.  ...  Consider an autonomous discrete time dynamical system in R p (3.1) x + = (x), x ∈ R p can be used to define a Koopman operator, U , that acts on functions on the statespace: [14, 26] The space under  ... 
arXiv:2105.03607v2 fatcat:sdwgapaz2naf3br4u3o35szrbi
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