Invariant Sets in Quasiperiodically Forced Dynamical Systems

Susuki, Yoshihiko; Mezić, Igor

doi:10.1137/18m1193529

Cited by 4 publications

(2 citation statements)

References 42 publications

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“…can be treated in an analogous fashion. Assuming that the driving dynamics evolves on a compact state space, or otherwise is measurable with a well-defined invariant measure, is sufficient to justify computation of eigenfunctions using ergodic averages (2.30) for time-periodic systems and for systems driven by quasiperiodic dynamics [56,323].…”

Section: Koopman Operators For Nonautonomous and Stochastic Dynamicsmentioning

confidence: 99%

Modern Koopman Theory for Dynamical Systems

Brunton¹,

Budišić²,

Kaiser³

et al. 2021

Preprint

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The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinitedimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the dynamics appear approximately linear remains a central open challenge. 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 in terms of measurements, making it ideal for leveraging big-data and machine learning techniques, and 3) simple, yet powerful numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and extended to reduce Koopman theory to practice in real-world applications. In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications. We also discuss key advances and challenges in the rapidly growing field of machine learning that are likely to drive future developments and significantly transform the theoretical landscape of dynamical systems.

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Section: Koopman Operators For Nonautonomous and Stochastic Dynamicsmentioning

confidence: 99%

Modern Koopman Theory for Dynamical Systems

Brunton¹,

Budišić²,

Kaiser³

et al. 2021

Preprint

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show abstract

“…To establish a connection between RKHSs and nonlinear dynamical systems the following operator was introduced, which was inspired by the study of occupation measures and related concepts (cf. [27,2,34,21,57,22,4,5,8,28,29,36,38,61]). Definition 3.1.…”

Section: Prerequisitesmentioning

confidence: 99%

Occupation Kernel Hilbert Spaces for Fractional Order Liouville Operators and Dynamic Mode Decomposition

Rosenfeld¹,

Russo²,

Li³

2021

Preprint

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This manuscript introduces a space of functions, termed occupation kernel Hilbert space (OKHS), that operate on collections of signals rather than real or complex functions. To support this new definition, an explicit class of OKHSs is given through the consideration of a reproducing kernel Hilbert space (RKHS). This space enables the definition of nonlocal operators, such as fractional order Liouville operators, as well as spectral decomposition methods for corresponding fractional order dynamical systems. In this manuscript, a fractional order DMD routine is presented, and the details of the finite rank representations are given. Significantly, despite the added theoretical content through the OKHS formulation, the resultant computations only differ slightly from that of occupation kernel DMD methods for integer order systems posed over RKHSs.

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Strange nonchaotic attractors in a class of quasiperiodically forced piecewise smooth systems

Duan,

Wei,

et al. 2024

Nonlinear Dyn

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Invariant Sets in Quasiperiodically Forced Dynamical Systems

Cited by 4 publications

References 42 publications

Modern Koopman Theory for Dynamical Systems

Modern Koopman Theory for Dynamical Systems

Occupation Kernel Hilbert Spaces for Fractional Order Liouville Operators and Dynamic Mode Decomposition

Strange nonchaotic attractors in a class of quasiperiodically forced piecewise smooth systems

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