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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: closed (5 April 2023) | Viewed by 34672

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Special Issue Editors

Prof. Dr. ALEXANDRE MAUROY

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Guest Editor

Namur Center for Complex Systems, Department of Mathematics, University of Namur, 5000 Namur, Belgium
Interests: Dynamical Systems; Control Theory; Operator Theory; Networks

Prof. Dr. Yoshihiko SUSUKI

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Guest Editor

Osaka Prefecture University, Kyoto 606-8501, Japan
Interests: Dynamical Systems; Energy Technology; Data Science; Control

Prof. Dr. Igor Mezic

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Mechanical Engineering, University of California, Santa Barbara 93106-5070, CA, USA
Interests: Dynamical Systems; Complexity; Artificial Intelligence; Network Security; operator theory

Special Issue Information

Dear Colleagues,

The classical approach to dynamical systems relies on the pointwise description of trajectories in the state space. However, there is an alternative global description in terms of functions defined on the state space (e.g., observables, densities), whose evolution is governed by linear operators. This operator-theoretic framework is typically conducive to linear, spectral analysis and is amenable to data-driven methods.

This Special Issue will collect high-quality research or review papers on recent developments of operator-theoretic methods in the broad context of dynamical systems theory. The main focus is on, but not limited to, the framework of the dual Koopman (or composition) operator and Perron–Frobenius (or transfer) operator. We await theoretical contributions highlighting connections between the operator-theoretic description of dynamical systems (e.g., spectral properties) and their geometric properties in the state space (e.g., attractors, coherent structures). Contributions proposing novel (possibly data-driven) numerical methods to compute finite-dimensional approximations of the operators, with possible connections to or use of dynamic mode decomposition, finite element/volume methods, or machine learning will also be appreciated. Finally, we anticipate cutting-edge applications of operator-theoretic methods to specific domains, such as control theory, engineering, fluid dynamics, computational biology, and network science, to list a few.

Prof. Dr. ALEXANDRE MAUROY
Prof. Dr. Yoshihiko SUSUKI
Prof. Dr. Igor Mezic
Guest Editors

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Keywords

operator theory
dynamic systems
Koopman operator
Perron–Frobenius operator
composition operator
transfer operator
dynamic mode decomposition

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Published Papers (12 papers)

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Research

18 pages, 336 KiB

Open AccessArticle

On the Joint A-Numerical Radius of Operators and Related Inequalities

by Najla Altwaijry, Silvestru Sever Dragomir and Kais Feki

Mathematics 2023, 11(10), 2293; https://doi.org/10.3390/math11102293 - 15 May 2023

Cited by 4 | Viewed by 1232

Abstract

In this paper, we study p-tuples of bounded linear operators on a complex Hilbert space with adjoint operators defined with respect to a non-zero positive operator A. Our main objective is to investigate the joint A-numerical radius of the p-tuple.We established several upper bounds for it, some of which extend and improve upon a previous work of the second author. Additionally, we provide several sharp inequalities involving the classical A-numerical radius and the A-seminorm of semi-Hilbert space operators as applications of our results. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

28 pages, 396 KiB

Open AccessEditor’s ChoiceArticle

On Numerical Approximations of the Koopman Operator

by Igor Mezić

Mathematics 2022, 10(7), 1180; https://doi.org/10.3390/math10071180 - 5 Apr 2022

Cited by 18 | Viewed by 4185

Abstract

We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode Decomposition-type methods in the context of Finite Section theory of infinite dimensional operators, and provide an example of a mixing map for which the finite section method fails. Under assumptions on the underlying dynamics, we provide the first result on the convergence rate under sample size increase in the finite-section approximation. We study the error in the Krylov subspace version of the finite section method and prove convergence in pseudospectral sense for operators with pure point spectrum. Since Krylov sequence-based approximations can mitigate the curse of dimensionality, this result indicates that they may also have low spectral error without an exponential-in-dimension increase in the number of functions needed. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

25 pages, 867 KiB

Open AccessArticle

Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators

by Jamiree Harrison and Enoch Yeung

Mathematics 2021, 9(23), 3133; https://doi.org/10.3390/math9233133 - 5 Dec 2021

Cited by 5 | Viewed by 3212

Abstract

The genetic toggle switch is a well known model in synthetic biology that represents the dynamic interactions between two genes that repress each other. The mathematical models for the genetic toggle switch that currently exist have been useful in describing circuit dynamics in rapidly dividing cells, assuming fixed or time-invariant kinetic rates. There is a growing interest in being able to model and extend synthetic biological function for growth conditions such as stationary phase or during nutrient starvation. As cells transition from one growth phase to another, kinetic rates become time-varying parameters. In this paper, we propose a novel class of parameter varying nonlinear models that can be used to describe the dynamics of genetic circuits, including the toggle switch, as they transition from different phases of growth. We show that there exists unique solutions for this class of systems, as well as for a class of systems that incorporates the microbial phenomena of quorum sensing. Further, we show that the domain of these systems, which is the positive orthant, is positively invariant. We also showcase a theoretical control strategy for these systems that would grant asymptotic monostability of a desired fixed point. We then take the general form of these systems and analyze their stability properties through the framework of time-varying Koopman operator theory. A necessary condition for asymptotic stability is also provided as well as a sufficient condition for instability. A Koopman control strategy for the system is also proposed, as well as an analogous discrete time-varying Koopman framework for applications with regularly sampled measurements. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

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17 pages, 11321 KiB

Open AccessArticle

Randomized Projection Learning Method for Dynamic Mode Decomposition

by Sudam Surasinghe and Erik M. Bollt

Mathematics 2021, 9(21), 2803; https://doi.org/10.3390/math9212803 - 4 Nov 2021

Cited by 4 | Viewed by 2310

Abstract

A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on a projected space. In the spirit of Johnson–Lindenstrauss lemma, we will use a random projection to estimate the DMD modes in a reduced dimensional space. In practical applications, snapshots are in a high-dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is expensive, so our main computational goal is to estimate the eigenvalue and eigenvectors of the DMD operator in a projected domain. We generalize the current algorithm to estimate a projected DMD operator. We focus on a powerful and simple random projection algorithm that will reduce the computational and storage costs. While, clearly, a random projection simplifies the algorithmic complexity of a detailed optimal projection, as we will show, the results can generally be excellent, nonetheless, and the quality could be understood through a well-developed theory of random projections. We will demonstrate that modes could be calculated for a low cost by the projected data with sufficient dimension. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

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20 pages, 489 KiB

Open AccessEditor’s ChoiceArticle

Is the Finite-Time Lyapunov Exponent Field a Koopman Eigenfunction?

by Erik M. Bollt and Shane D. Ross

Mathematics 2021, 9(21), 2731; https://doi.org/10.3390/math9212731 - 28 Oct 2021

Cited by 1 | Viewed by 3593

Abstract

This work serves as a bridge between two approaches to analysis of dynamical systems: the local, geometric analysis, and the global operator theoretic Koopman analysis. We explicitly construct vector fields where the instantaneous Lyapunov exponent field is a Koopman eigenfunction. Restricting ourselves to polynomial vector fields to make this construction easier, we find that such vector fields do exist, and we explore whether such vector fields have a special structure, thus making a link between the geometric theory and the transfer operator theory. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

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14 pages, 874 KiB

Open AccessArticle

Koopman Operator Framework for Spectral Analysis and Identification of Infinite-Dimensional Systems

by Alexandre Mauroy

Mathematics 2021, 9(19), 2495; https://doi.org/10.3390/math9192495 - 5 Oct 2021

Cited by 3 | Viewed by 2988

Abstract

We consider the Koopman operator theory in the context of nonlinear infinite-dimensional systems, where the operator is defined over a space of bounded continuous functionals. The properties of the Koopman semigroup are described and a finite-dimensional projection of the semigroup is proposed, which provides a linear finite-dimensional approximation of the underlying infinite-dimensional dynamics. This approximation is used to obtain spectral properties from the data, a method which can be seen as a generalization of the Extended Dynamic Mode Decomposition for infinite-dimensional systems. Finally, we exploit the proposed framework to identify (a finite-dimensional approximation of) the Lie generator associated with the Koopman semigroup. This approach yields a linear method for nonlinear PDE identification, which is complemented with theoretical convergence results. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

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12 pages, 764 KiB

Open AccessArticle

A Convex Data-Driven Approach for Nonlinear Control Synthesis

by Hyungjin Choi, Umesh Vaidya and Yongxin Chen

Mathematics 2021, 9(19), 2445; https://doi.org/10.3390/math9192445 - 1 Oct 2021

Cited by 8 | Viewed by 2048

Abstract

We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

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18 pages, 1672 KiB

Open AccessArticle

Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory

by Yuzuru Kato, Jinjie Zhu, Wataru Kurebayashi and Hiroya Nakao

Mathematics 2021, 9(18), 2188; https://doi.org/10.3390/math9182188 - 7 Sep 2021

Cited by 9 | Viewed by 3175

Abstract

The asymptotic phase is a fundamental quantity for the analysis of deterministic limit-cycle oscillators, and generalized definitions of the asymptotic phase for stochastic oscillators have also been proposed. In this article, we show that the asymptotic phase and also amplitude can be defined for classical and semiclassical stochastic oscillators in a natural and unified manner by using the eigenfunctions of the Koopman operator of the system. We show that the proposed definition gives appropriate values of the phase and amplitude for strongly stochastic limit-cycle oscillators, excitable systems undergoing noise-induced oscillations, and also for quantum limit-cycle oscillators in the semiclassical regime. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

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29 pages, 8693 KiB

Open AccessArticle

Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method

by Zlatko Drmač, Igor Mezić and Ryan Mohr

Mathematics 2021, 9(17), 2075; https://doi.org/10.3390/math9172075 - 27 Aug 2021

Cited by 4 | Viewed by 2611

Abstract

Inferring the latent structure of complex nonlinear dynamical systems in a data driven setting is a challenging mathematical problem with an ever increasing spectrum of applications in sciences and engineering. Koopman operator-based linearization provides a powerful framework that is suitable for identification of nonlinear systems in various scenarios. A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup. This elegant and mathematically appealing approach has good analytical (convergence) properties, but numerical experiments show that software implementation of the method has certain limitations. More precisely, with the increased dimension that guarantees theoretically better approximation and ultimate convergence, the numerical implementation may become unstable and it may even break down. The main sources of numerical difficulties are the computations of the matrix representation of the compressed Koopman operator and its logarithm. This paper addresses the subtle numerical details and proposes a new implementation algorithm that alleviates these problems. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

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15 pages, 833 KiB

Open AccessArticle

Trigonometric Embeddings in Polynomial Extended Mode Decomposition—Experimental Application to an Inverted Pendulum

by Camilo Garcia-Tenorio, Gilles Delansnay, Eduardo Mojica-Nava and Alain Vande Wouwer

Mathematics 2021, 9(10), 1119; https://doi.org/10.3390/math9101119 - 15 May 2021

Cited by 5 | Viewed by 2244

Abstract

The extended dynamic mode decomposition algorithm is a tool for accurately approximating the point spectrum of the Koopman operator. This algorithm provides an approximate linear expansion of non-linear discrete-time systems, which can be useful for system analysis and controller design. The accuracy of this algorithm depends heavily on the availability of a set of basis functions that provide the ability to capture the nonlinear dynamics of the underlying system. Recently, the use of orthogonal polynomials, along with reduction techniques for the dimension and maximum order of the polynomial basis, have been successfully used to approximate nonlinear systems with the additional benefit of using smaller datasets. This paper expands the current methods for selecting the set of observables for nonlinear systems with periodic behavior, which is prone to a representation in terms of trigonometric functions. The benefit of working with orthogonal polynomials is preserved by embedding the trigonometric functions into the orthogonal basis. The algorithm is illustrated with the data-driven modelling of an inverted pendulum in simulation and real-life experiments. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

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19 pages, 13300 KiB

Open AccessArticle

Robust Mode Analysis

by Gemunu H. Gunaratne and Sukesh Roy

Mathematics 2021, 9(9), 1057; https://doi.org/10.3390/math9091057 - 8 May 2021

Cited by 2 | Viewed by 2240

Abstract

In this paper, we introduce a model-free algorithm, robust mode analysis (RMA), to extract primary constituents in a fluid or reacting flow directly from high-frequency, high-resolution experimental data. It is expected to be particularly useful in studying strongly driven flows, where nonlinearities can induce chaotic and irregular dynamics. The lack of precise governing equations and the absence of symmetries or other simplifying constraints in realistic configurations preclude the derivation of analytical solutions for these systems; the presence of flow structures over a wide range of scales handicaps finding their numerical solutions. Thus, the need for direct analysis of experimental data is reinforced. RMA is predicated on the assumption that primary flow constituents are common in multiple, nominally identical realizations of an experiment. Their search relies on the identification of common dynamic modes in the experiments, the commonality established via proximity of the eigenvalues and eigenfunctions. Robust flow constituents are then constructed by combining common dynamic modes that flow at the same rate. We illustrate RMA using reacting flows behind a symmetric bluff body. Two robust constituents, whose signatures resemble symmetric and von Karman vortex shedding, are identified. It is shown how RMA can be implemented via extended dynamic mode decomposition in flow configurations interrogated with a small number of time-series. This approach may prove useful in analyzing changes in flow patterns in engines and propulsion systems equipped with sturdy arrays of pressure transducers or thermocouples. Finally, an analysis of high Reynolds number jet flows suggests that tests of statistical characterizations in turbulent flows may best be done using non-robust components of the flow. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

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18 pages, 585 KiB

Open AccessArticle

Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design

by Keita Hara and Masaki Inoue

Mathematics 2021, 9(9), 949; https://doi.org/10.3390/math9090949 - 23 Apr 2021

Cited by 3 | Viewed by 2314

Abstract

In this paper, we address the data-driven modeling of a nonlinear dynamical system while incorporating a priori information. The nonlinear system is described using the Koopman operator, which is a linear operator defined on a lifted infinite-dimensional state-space. Assuming that the

L_{2}

[...] Read more.

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gain of the system is known, the data-driven finite-dimensional approximation of the operator while preserving information about the gain, namely

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gain-preserving data-driven modeling, is formulated. Then, its computationally efficient solution method is presented. An application of the modeling method to feedback controller design is also presented. Aiming for robust stabilization using data-driven control under a poor training dataset, we address the following two modeling problems: (1) Forward modeling: the data-driven modeling is applied to the operating data of a plant system to derive the plant model; (2) Backward modeling:

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gain-preserving data-driven modeling is applied to the same data to derive an inverse model of the plant system. Then, a feedback controller composed of the plant and inverse models is created based on internal model control, and it robustly stabilizes the plant system. A design demonstration of the data-driven controller is provided using a numerical experiment. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

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