SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1133-1158, June 2024. Abstract.We study the dynamics of a swarmalator model with higher harmonic phase coupling. We analyze stability, bifurcation, and structural properties of several novel attracting states, including the formation of spatial clusters with distinct phases, and single spatial clusters with a small number of distinct phases. We use mean-field (centroid) dynamics to analytically determine intercluster distance. We also find states with two large clusters along with a small number of swarmalators that are trapped between the two clusters and vacillate (waver) between them. In the case of a single vacillator we use a mean-field reduction to reduce the dynamics to two dimensions, which enables a detailed bifurcation analysis. We show excellent agreement between our reduced two-dimensional model and the dynamics and bifurcations of the full swarmalator model.
{"title":"Swarmalators with Higher Harmonic Coupling: Clustering and Vacillating","authors":"Lauren D. Smith","doi":"10.1137/23m1606460","DOIUrl":"https://doi.org/10.1137/23m1606460","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1133-1158, June 2024. <br/> Abstract.We study the dynamics of a swarmalator model with higher harmonic phase coupling. We analyze stability, bifurcation, and structural properties of several novel attracting states, including the formation of spatial clusters with distinct phases, and single spatial clusters with a small number of distinct phases. We use mean-field (centroid) dynamics to analytically determine intercluster distance. We also find states with two large clusters along with a small number of swarmalators that are trapped between the two clusters and vacillate (waver) between them. In the case of a single vacillator we use a mean-field reduction to reduce the dynamics to two dimensions, which enables a detailed bifurcation analysis. We show excellent agreement between our reduced two-dimensional model and the dynamics and bifurcations of the full swarmalator model.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140834228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1108-1132, June 2024. Abstract.The immune response is a crucial factor in controlling HIV infection. However, oxidative stress poses a significant challenge to the HIV-specific immune response, compromising the body’s ability to control viral replication. In this paper, we develop an HIV infection model to investigate the impact of immune impairment on virus dynamics. We derive the basic reproduction number ([math]) and threshold ([math]). Utilizing the antioxidant parameter as a bifurcation parameter, we establish that the system exhibits saddle-node bifurcation backward and forward bifurcations. Specifically, when [math], the virus will rebound if the antioxidant parameter falls below the post-treatment control threshold. Conversely, when the antioxidant parameter exceeds the elite control threshold, the virus remains under elite control. The region between the two thresholds represents a bistable interval. These results can explain why some HIV-infected patients experience rapid viral rebound after treatment cessation while others achieve post-treatment control for a longer time.
{"title":"Bistability of an HIV Model with Immune Impairment","authors":"Shaoli Wang, Tengfei Wang, Fei Xu, Libin Rong","doi":"10.1137/23m1596004","DOIUrl":"https://doi.org/10.1137/23m1596004","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1108-1132, June 2024. <br/> Abstract.The immune response is a crucial factor in controlling HIV infection. However, oxidative stress poses a significant challenge to the HIV-specific immune response, compromising the body’s ability to control viral replication. In this paper, we develop an HIV infection model to investigate the impact of immune impairment on virus dynamics. We derive the basic reproduction number ([math]) and threshold ([math]). Utilizing the antioxidant parameter as a bifurcation parameter, we establish that the system exhibits saddle-node bifurcation backward and forward bifurcations. Specifically, when [math], the virus will rebound if the antioxidant parameter falls below the post-treatment control threshold. Conversely, when the antioxidant parameter exceeds the elite control threshold, the virus remains under elite control. The region between the two thresholds represents a bistable interval. These results can explain why some HIV-infected patients experience rapid viral rebound after treatment cessation while others achieve post-treatment control for a longer time.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140834542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1090-1107, June 2024. Abstract.It has long been known that the excitation of fast motion in certain two-scale dynamical systems is linked to the singularity structure in complex time of the slow variables. We demonstrate that, in the context of a fast harmonic oscillator forced by one component of the Lorenz 1963 model, this principle can be used to construct time-discrete surrogate models by numerically extracting approximate locations and residues of complex poles via adaptive Antoulas–Anderson (AAA) rational interpolation and feeding this information into the known “connection formula” to compute the resulting fast amplitude. Despite small but nonnegligible local errors, the surrogate model maintains excellent accuracy over very long times. In addition, we observe that the long-time behavior of fast energy offers a continuous-time analogue of Gottwald and Melbourne’s 2004 “0–1 test for chaos”; that is, the asymptotic growth rate of the energy in the oscillator can discern whether or not the forcing function is chaotic.
{"title":"Deterministic and Stochastic Surrogate Models for a Slowly Driven Fast Oscillator","authors":"Marcel Oliver, Marc A. Tiofack Kenfack","doi":"10.1137/23m1602176","DOIUrl":"https://doi.org/10.1137/23m1602176","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1090-1107, June 2024. <br/> Abstract.It has long been known that the excitation of fast motion in certain two-scale dynamical systems is linked to the singularity structure in complex time of the slow variables. We demonstrate that, in the context of a fast harmonic oscillator forced by one component of the Lorenz 1963 model, this principle can be used to construct time-discrete surrogate models by numerically extracting approximate locations and residues of complex poles via adaptive Antoulas–Anderson (AAA) rational interpolation and feeding this information into the known “connection formula” to compute the resulting fast amplitude. Despite small but nonnegligible local errors, the surrogate model maintains excellent accuracy over very long times. In addition, we observe that the long-time behavior of fast energy offers a continuous-time analogue of Gottwald and Melbourne’s 2004 “0–1 test for chaos”; that is, the asymptotic growth rate of the energy in the oscillator can discern whether or not the forcing function is chaotic.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140812982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1052-1089, June 2024. Abstract.Spectral subspaces of a linear dynamical system identify a large class of invariant structures that highlight/isolate the dynamics associated to select subsets of the spectrum. The corresponding notion for nonlinear systems is that of spectral submanifolds—manifolds invariant under the full nonlinear dynamics that are determined by their tangency to spectral subspaces of the linearized system. In light of the recently emerged interest in their use as tools in model reduction, we propose an extension of the relevant theory to the realm of fluid dynamics. We show the existence of a large (and the most pertinent) subclass of spectral submanifolds and foliations—describing the behavior of nearby trajectories—about fixed points and periodic orbits of the Navier–Stokes equations. Their uniqueness and smoothness properties are discussed in detail, due to their significance from the perspective of model reduction. The machinery is then put to work via a numerical algorithm developed along the lines of the parameterization method, which computes the desired manifolds as power series expansions. Results are shown within the context of two-dimensional channel flows.
{"title":"Spectral Submanifolds of the Navier–Stokes Equations","authors":"Gergely Buza","doi":"10.1137/23m154858x","DOIUrl":"https://doi.org/10.1137/23m154858x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1052-1089, June 2024. <br/> Abstract.Spectral subspaces of a linear dynamical system identify a large class of invariant structures that highlight/isolate the dynamics associated to select subsets of the spectrum. The corresponding notion for nonlinear systems is that of spectral submanifolds—manifolds invariant under the full nonlinear dynamics that are determined by their tangency to spectral subspaces of the linearized system. In light of the recently emerged interest in their use as tools in model reduction, we propose an extension of the relevant theory to the realm of fluid dynamics. We show the existence of a large (and the most pertinent) subclass of spectral submanifolds and foliations—describing the behavior of nearby trajectories—about fixed points and periodic orbits of the Navier–Stokes equations. Their uniqueness and smoothness properties are discussed in detail, due to their significance from the perspective of model reduction. The machinery is then put to work via a numerical algorithm developed along the lines of the parameterization method, which computes the desired manifolds as power series expansions. Results are shown within the context of two-dimensional channel flows.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1017-1051, June 2024. Abstract.In this paper, we give an in-depth error analysis for surrogate models generated by a variant of the Sparse Identification of Nonlinear Dynamics (SINDy) method. We start with an overview of a variety of nonlinear system identification techniques, namely SINDy, weak-SINDy, and the occupation kernel method. Under the assumption that the dynamics are a finite linear combination of a set of basis functions, these methods establish a linear system to recover coefficients. We illuminate the structural similarities between these techniques and establish a projection property for the weak-SINDy technique. Following the overview, we analyze the error of surrogate models generated by a simplified version of weak-SINDy. In particular, under the assumption of boundedness of a composition operator given by the solution, we show that (i) the surrogate dynamics converges towards the true dynamics and (ii) the solution of the surrogate model is reasonably close to the true solution. Finally, as an application, we discuss the use of a combination of weak-SINDy surrogate modeling and proper orthogonal decomposition (POD) to build a surrogate model for partial differential equations (PDEs).
{"title":"Convergence of Weak-SINDy Surrogate Models","authors":"Benjamin P. Russo, M. Paul Laiu","doi":"10.1137/22m1526782","DOIUrl":"https://doi.org/10.1137/22m1526782","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1017-1051, June 2024. <br/>Abstract.In this paper, we give an in-depth error analysis for surrogate models generated by a variant of the Sparse Identification of Nonlinear Dynamics (SINDy) method. We start with an overview of a variety of nonlinear system identification techniques, namely SINDy, weak-SINDy, and the occupation kernel method. Under the assumption that the dynamics are a finite linear combination of a set of basis functions, these methods establish a linear system to recover coefficients. We illuminate the structural similarities between these techniques and establish a projection property for the weak-SINDy technique. Following the overview, we analyze the error of surrogate models generated by a simplified version of weak-SINDy. In particular, under the assumption of boundedness of a composition operator given by the solution, we show that (i) the surrogate dynamics converges towards the true dynamics and (ii) the solution of the surrogate model is reasonably close to the true solution. Finally, as an application, we discuss the use of a combination of weak-SINDy surrogate modeling and proper orthogonal decomposition (POD) to build a surrogate model for partial differential equations (PDEs).","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Andrey Bychkov, Opal Issan, Gleb Pogudin, Boris Kramer
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 982-1016, March 2024. Abstract. Quadratization of polynomial and nonpolynomial systems of ordinary differential equations (ODEs) is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling, and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, and control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms, and software capabilities for quadratization of nonautonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semidiscretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the QBee software towards both nonautonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages.
{"title":"Exact and Optimal Quadratization of Nonlinear Finite-Dimensional Nonautonomous Dynamical Systems","authors":"Andrey Bychkov, Opal Issan, Gleb Pogudin, Boris Kramer","doi":"10.1137/23m1561129","DOIUrl":"https://doi.org/10.1137/23m1561129","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 982-1016, March 2024. <br/> Abstract. Quadratization of polynomial and nonpolynomial systems of ordinary differential equations (ODEs) is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling, and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, and control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms, and software capabilities for quadratization of nonautonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semidiscretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the QBee software towards both nonautonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 961-981, March 2024. Abstract. This work is devoted to investigating the noise-induced rare transition of periodically driven systems. The maximum likelihood paths (MLPs) are often sought, in order to reveal the transition mechanism. We show that MLPs between metastable periodic states could persist to a small nonautonomous forcing under appropriate conditions. Furthermore, we obtain a closed-form explicit expression for approximating the transition rate change. They are obtained based on standard perturbation techniques for the Euler–Lagrange equation, the Melnikov theory, as well as a linear-theory calculation. Our methods indicate a route for a detailed understanding for the interaction between periodic forcing and noise in rather general systems.
{"title":"Small-Noise-Induced Metastable Transition of Periodically Perturbed Systems","authors":"Ying Chao, Jinqiao Duan, Pingyuan Wei","doi":"10.1137/23m1567308","DOIUrl":"https://doi.org/10.1137/23m1567308","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 961-981, March 2024. <br/> Abstract. This work is devoted to investigating the noise-induced rare transition of periodically driven systems. The maximum likelihood paths (MLPs) are often sought, in order to reveal the transition mechanism. We show that MLPs between metastable periodic states could persist to a small nonautonomous forcing under appropriate conditions. Furthermore, we obtain a closed-form explicit expression for approximating the transition rate change. They are obtained based on standard perturbation techniques for the Euler–Lagrange equation, the Melnikov theory, as well as a linear-theory calculation. Our methods indicate a route for a detailed understanding for the interaction between periodic forcing and noise in rather general systems.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 924-960, March 2024. Abstract. Nonlinear ODEs can rarely be solved analytically. Koopman operator theory provides a way to solve two-dimensional nonlinear systems, under suitable restrictions, by mapping nonlinear dynamics to a linear space using Koopman eigenfunctions. Unfortunately, finding such eigenfunctions is difficult. We introduce a method for constructing Koopman eigenfunctions from a two-dimensional nonlinear ODE’s one-dimensional invariant manifolds. This method, when successful, allows us to find analytical solutions for autonomous, nonlinear systems. Previous data-driven methods have used Koopman theory to construct local Koopman eigenfunction approximations valid in different regions of phase space; our method finds analytic Koopman eigenfunctions that are exact and globally valid. We demonstrate our Koopman method of solving nonlinear systems on one-dimensional and two-dimensional ODEs. The nonlinear examples considered have simple expressions for their codimension-1 invariant manifolds which produce tractable analytical solutions. Thus our method allows for the construction of analytical solutions for previously unsolved ODEs. It also highlights the connection between invariant manifolds and eigenfunctions in nonlinear ODEs and presents avenues for extending this method to solve more nonlinear systems.
{"title":"Solving Nonlinear Ordinary Differential Equations Using the Invariant Manifolds and Koopman Eigenfunctions","authors":"Megan Morrison, J. Nathan Kutz","doi":"10.1137/22m1516622","DOIUrl":"https://doi.org/10.1137/22m1516622","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 924-960, March 2024. <br/> Abstract. Nonlinear ODEs can rarely be solved analytically. Koopman operator theory provides a way to solve two-dimensional nonlinear systems, under suitable restrictions, by mapping nonlinear dynamics to a linear space using Koopman eigenfunctions. Unfortunately, finding such eigenfunctions is difficult. We introduce a method for constructing Koopman eigenfunctions from a two-dimensional nonlinear ODE’s one-dimensional invariant manifolds. This method, when successful, allows us to find analytical solutions for autonomous, nonlinear systems. Previous data-driven methods have used Koopman theory to construct local Koopman eigenfunction approximations valid in different regions of phase space; our method finds analytic Koopman eigenfunctions that are exact and globally valid. We demonstrate our Koopman method of solving nonlinear systems on one-dimensional and two-dimensional ODEs. The nonlinear examples considered have simple expressions for their codimension-1 invariant manifolds which produce tractable analytical solutions. Thus our method allows for the construction of analytical solutions for previously unsolved ODEs. It also highlights the connection between invariant manifolds and eigenfunctions in nonlinear ODEs and presents avenues for extending this method to solve more nonlinear systems.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 885-923, March 2024. Abstract.Data-driven models for nonlinear dynamical systems based on approximating the underlying Koopman operator or generator have proven to be successful tools for forecasting, feature learning, state estimation, and control. It has become well known that the Koopman generators for control-affine systems also have affine dependence on the input, leading to convenient finite-dimensional bilinear approximations of the dynamics. Yet there are still two main obstacles that limit the scope of current approaches for approximating the Koopman generators of systems with actuation. First, the performance of existing methods depends heavily on the choice of basis functions over which the Koopman generator is to be approximated; and there is currently no universal way to choose them for systems that are not measure preserving. Second, if we do not observe the full state, then it becomes necessary to account for the dependence of the output time series on the sequence of supplied inputs when constructing observables to approximate Koopman operators. To address these issues, we write the dynamics of observables governed by the Koopman generator as a bilinear hidden Markov model and determine the model parameters using the expectation-maximization algorithm. The E step involves a standard Kalman filter and smoother, while the M step resembles control-affine dynamic mode decomposition for the generator. We demonstrate the performance of this method on three examples, including recovery of a finite-dimensional Koopman-invariant subspace for an actuated system with a slow manifold; estimation of Koopman eigenfunctions for the unforced Duffing equation; and model-predictive control of a fluidic pinball system based only on noisy observations of lift and drag.
{"title":"Learning Bilinear Models of Actuated Koopman Generators from Partially Observed Trajectories","authors":"Samuel Otto, Sebastian Peitz, Clarence Rowley","doi":"10.1137/22m1523601","DOIUrl":"https://doi.org/10.1137/22m1523601","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 885-923, March 2024. <br/> Abstract.Data-driven models for nonlinear dynamical systems based on approximating the underlying Koopman operator or generator have proven to be successful tools for forecasting, feature learning, state estimation, and control. It has become well known that the Koopman generators for control-affine systems also have affine dependence on the input, leading to convenient finite-dimensional bilinear approximations of the dynamics. Yet there are still two main obstacles that limit the scope of current approaches for approximating the Koopman generators of systems with actuation. First, the performance of existing methods depends heavily on the choice of basis functions over which the Koopman generator is to be approximated; and there is currently no universal way to choose them for systems that are not measure preserving. Second, if we do not observe the full state, then it becomes necessary to account for the dependence of the output time series on the sequence of supplied inputs when constructing observables to approximate Koopman operators. To address these issues, we write the dynamics of observables governed by the Koopman generator as a bilinear hidden Markov model and determine the model parameters using the expectation-maximization algorithm. The E step involves a standard Kalman filter and smoother, while the M step resembles control-affine dynamic mode decomposition for the generator. We demonstrate the performance of this method on three examples, including recovery of a finite-dimensional Koopman-invariant subspace for an actuated system with a slow manifold; estimation of Koopman eigenfunctions for the unforced Duffing equation; and model-predictive control of a fluidic pinball system based only on noisy observations of lift and drag.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Katherine Morrison, Anda Degeratu, Vladimir Itskov, Carina Curto
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 855-884, March 2024. Abstract.Threshold-linear networks consist of simple units interacting in the presence of a threshold nonlinearity. Competitive threshold-linear networks have long been known to exhibit multistability, where the activity of the network settles into one of potentially many steady states. In this work, we find conditions that guarantee the absence of steady states, while maintaining bounded activity. These conditions lead us to define a combinatorial family of competitive threshold-linear networks, parametrized by a simple directed graph. By exploring this family, we discover that threshold-linear networks are capable of displaying a surprisingly rich variety of nonlinear dynamics, including limit cycles, quasi-periodic attractors, and chaos. In particular, several types of nonlinear behaviors can co-exist in the same network. Our mathematical results also enable us to engineer networks with multiple dynamic patterns. Taken together, these theoretical and computational findings suggest that threshold-linear networks may be a valuable tool for understanding the relationship between network connectivity and emergent dynamics.
{"title":"Diversity of Emergent Dynamics in Competitive Threshold-Linear Networks","authors":"Katherine Morrison, Anda Degeratu, Vladimir Itskov, Carina Curto","doi":"10.1137/22m1541666","DOIUrl":"https://doi.org/10.1137/22m1541666","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 855-884, March 2024. <br/> Abstract.Threshold-linear networks consist of simple units interacting in the presence of a threshold nonlinearity. Competitive threshold-linear networks have long been known to exhibit multistability, where the activity of the network settles into one of potentially many steady states. In this work, we find conditions that guarantee the absence of steady states, while maintaining bounded activity. These conditions lead us to define a combinatorial family of competitive threshold-linear networks, parametrized by a simple directed graph. By exploring this family, we discover that threshold-linear networks are capable of displaying a surprisingly rich variety of nonlinear dynamics, including limit cycles, quasi-periodic attractors, and chaos. In particular, several types of nonlinear behaviors can co-exist in the same network. Our mathematical results also enable us to engineer networks with multiple dynamic patterns. Taken together, these theoretical and computational findings suggest that threshold-linear networks may be a valuable tool for understanding the relationship between network connectivity and emergent dynamics.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140128149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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