SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1345-1371, June 2024. Abstract.In a recent article by Guglielmi and Hairer [SIAM J. Appl. Dyn. Syst., 14 (2015), pp. 1454–1477], an analysis in the [math] limit was proposed of regularized discontinuous ODEs in codimension-2 switching domains; this was obtained by studying a certain 2-dimensional system describing the so-called hidden dynamics. In particular, the existence of a unique limit solution was not proved in all cases, a few of which were labeled as ambiguous, and it was not clear whether or not the ambiguity could be resolved. In this paper, we show that it cannot be resolved in general. A first contribution of this paper is an illustration of the dependence of the limit solution on the form of the switching function. Considering the parameter dependence in the ambiguous class of discontinuous systems, a second contribution is a bifurcation analysis, revealing a range of possible behaviors. Finally, we investigate the sensitivity of solutions in the transition from codimension-2 domains to codimension-3 when there is a limit cycle in the hidden dynamics.
{"title":"Nonuniqueness Phenomena in Discontinuous Dynamical Systems and Their Regularizations","authors":"Alessia Andò, Roderick Edwards, Nicola Guglielmi","doi":"10.1137/23m1587488","DOIUrl":"https://doi.org/10.1137/23m1587488","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1345-1371, June 2024. <br/> Abstract.In a recent article by Guglielmi and Hairer [SIAM J. Appl. Dyn. Syst., 14 (2015), pp. 1454–1477], an analysis in the [math] limit was proposed of regularized discontinuous ODEs in codimension-2 switching domains; this was obtained by studying a certain 2-dimensional system describing the so-called hidden dynamics. In particular, the existence of a unique limit solution was not proved in all cases, a few of which were labeled as ambiguous, and it was not clear whether or not the ambiguity could be resolved. In this paper, we show that it cannot be resolved in general. A first contribution of this paper is an illustration of the dependence of the limit solution on the form of the switching function. Considering the parameter dependence in the ambiguous class of discontinuous systems, a second contribution is a bifurcation analysis, revealing a range of possible behaviors. Finally, we investigate the sensitivity of solutions in the transition from codimension-2 domains to codimension-3 when there is a limit cycle in the hidden dynamics.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"3 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168382","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 1313-1344, June 2024. Abstract.This article investigates the nonlinear stability and dynamic bifurcation for the equations governing the Marangoni convection of two superimposed immiscible liquids subject to temperature gradient perpendicular to the plate. First, we obtain the critical value of the Marangoni number and verify the stability exchange principle by adopting a hybrid method that combines theoretical analysis and numerical calculations. Second, we employ the energy method, probing the nonlinear stability and establishing the nonlinear thresholds of the Marangoni number. Third, we apply the technique of center manifold reduction to reduce the corresponding infinite-dimensional model to finite-dimensional ODEs. According to the ODEs, we establish a dynamic bifurcation theorem with the transition number that determines the bifurcation type of the model. Finally, we determine the nondimensional transition number and present related temporal and flow patterns by performing careful numerical computation.
{"title":"On the Stability and Bifurcation of Marangoni Convection of Two Immiscible Liquids with a Nondeformable Interface","authors":"Chao Xing, Daozhi Han, Quan Wang","doi":"10.1137/23m1584174","DOIUrl":"https://doi.org/10.1137/23m1584174","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1313-1344, June 2024. <br/> Abstract.This article investigates the nonlinear stability and dynamic bifurcation for the equations governing the Marangoni convection of two superimposed immiscible liquids subject to temperature gradient perpendicular to the plate. First, we obtain the critical value of the Marangoni number and verify the stability exchange principle by adopting a hybrid method that combines theoretical analysis and numerical calculations. Second, we employ the energy method, probing the nonlinear stability and establishing the nonlinear thresholds of the Marangoni number. Third, we apply the technique of center manifold reduction to reduce the corresponding infinite-dimensional model to finite-dimensional ODEs. According to the ODEs, we establish a dynamic bifurcation theorem with the transition number that determines the bifurcation type of the model. Finally, we determine the nondimensional transition number and present related temporal and flow patterns by performing careful numerical computation.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"64 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141517555","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 1242-1271, June 2024. Abstract.This paper studies the dynamic interplay between inherent agent dynamics and velocity alignment in the ensemble of Cucker–Smale (CS) flocks under both fixed and switching interaction topologies. For the fixed topology network modeled by a rooted digraph, we provide sufficient frameworks for exponential convergence of flocking provided that the Lipschitz constant of nonlinear dynamics is less than a given bound in terms of initial state and system parameters. Similar to the CS system free of inherent dynamics, our results exhibit threshold phenomena depending on the quantity of collectivity in of the rooted digraph. For the case with switching topologies being rooted in a sequence of time-blocks, we present similar sufficient frameworks leading to convergence of flocking in terms of initial state and system parameters. Our results hold for both continuous time and discrete time. Numerical simulations are performed to illustrate our theoretical results.
{"title":"Flocks with Nonlinear Inherent Dynamics under Fixed and Switching Digraphs","authors":"Jiu-Gang Dong","doi":"10.1137/23m1574270","DOIUrl":"https://doi.org/10.1137/23m1574270","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1242-1271, June 2024. <br/> Abstract.This paper studies the dynamic interplay between inherent agent dynamics and velocity alignment in the ensemble of Cucker–Smale (CS) flocks under both fixed and switching interaction topologies. For the fixed topology network modeled by a rooted digraph, we provide sufficient frameworks for exponential convergence of flocking provided that the Lipschitz constant of nonlinear dynamics is less than a given bound in terms of initial state and system parameters. Similar to the CS system free of inherent dynamics, our results exhibit threshold phenomena depending on the quantity of collectivity in of the rooted digraph. For the case with switching topologies being rooted in a sequence of time-blocks, we present similar sufficient frameworks leading to convergence of flocking in terms of initial state and system parameters. Our results hold for both continuous time and discrete time. Numerical simulations are performed to illustrate our theoretical results.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"46 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925810","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 1199-1241, June 2024. Abstract.In recent years, classical epidemic models, which assume stationary behavior of individuals, have been extended to include an adaptive heterogeneous response of the population to the current state of the epidemic. However, it is widely accepted that human behavior can exhibit history-dependence as a consequence of learned experiences. This history-dependence is similar to the hysteresis effects that have been well studied in control theory. To illustrate the importance of history-dependence for epidemic theory, we study the dynamics of a variant of the SIRS model where individuals exhibit lazy-switch responses to prevalence dynamics. The resulting model, which includes the Preisach hysteresis operator, possesses a continuum of endemic equilibrium states characterized by different proportions of susceptible, infected, and recovered populations. We discuss stability properties of the endemic equilibrium set and relate them to the degree of heterogeneity of the adaptive response. In particular, our results suggest that heterogeneity promotes the convergence of the epidemic trajectory to an equilibrium state. Heterogeneity can be achieved by selective intervention policies targeting specific population groups. On the other hand, heterogeneous responses can lead to a higher peak of infection during the epidemic and a higher prevalence at the endemic equilibrium after the epidemic. These results support the argument that public health responses during the emergence of a new disease have long-term consequences for subsequent management efforts. The main mathematical contribution of this work is a new method of global stability analysis, which uses a family of Lyapunov functions corresponding to different branches of the hysteresis operator. It is well known that instability can result from switching from one flow to another even though each flow is stable (if the flows have different Lyapunov functions). We provide sufficient conditions for the convergence of trajectories to the equilibrium set for switched systems with the Preisach hysteresis operator.
{"title":"Global Stability of SIR Model with Heterogeneous Transmission Rate Modeled by the Preisach Operator","authors":"Ruofei Guan, Jana Kopfová, Dmitrii Rachinskii","doi":"10.1137/22m154274x","DOIUrl":"https://doi.org/10.1137/22m154274x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1199-1241, June 2024. <br/> Abstract.In recent years, classical epidemic models, which assume stationary behavior of individuals, have been extended to include an adaptive heterogeneous response of the population to the current state of the epidemic. However, it is widely accepted that human behavior can exhibit history-dependence as a consequence of learned experiences. This history-dependence is similar to the hysteresis effects that have been well studied in control theory. To illustrate the importance of history-dependence for epidemic theory, we study the dynamics of a variant of the SIRS model where individuals exhibit lazy-switch responses to prevalence dynamics. The resulting model, which includes the Preisach hysteresis operator, possesses a continuum of endemic equilibrium states characterized by different proportions of susceptible, infected, and recovered populations. We discuss stability properties of the endemic equilibrium set and relate them to the degree of heterogeneity of the adaptive response. In particular, our results suggest that heterogeneity promotes the convergence of the epidemic trajectory to an equilibrium state. Heterogeneity can be achieved by selective intervention policies targeting specific population groups. On the other hand, heterogeneous responses can lead to a higher peak of infection during the epidemic and a higher prevalence at the endemic equilibrium after the epidemic. These results support the argument that public health responses during the emergence of a new disease have long-term consequences for subsequent management efforts. The main mathematical contribution of this work is a new method of global stability analysis, which uses a family of Lyapunov functions corresponding to different branches of the hysteresis operator. It is well known that instability can result from switching from one flow to another even though each flow is stable (if the flows have different Lyapunov functions). We provide sufficient conditions for the convergence of trajectories to the equilibrium set for switched systems with the Preisach hysteresis operator.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"67 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942287","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 1159-1198, June 2024. Abstract.In this work we study the splitting distance of a rapidly perturbed pendulum [math] with [math] a [math]-periodic function and [math]. Systems of this kind undergo exponentially small splitting, and, when [math], it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided [math]. Our study focuses on the case [math], and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as [math]. On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency [math] in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with [math] where, for most [math], the perturbation satisfies [math]. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton–Jacobi formalism. The leading exponentially small term appears at order [math], where [math] is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.
{"title":"Splitting of Separatrices for Rapid Degenerate Perturbations of the Classical Pendulum","authors":"Inmaculada Baldomá, Tere M-Seara, Román Moreno","doi":"10.1137/23m1550992","DOIUrl":"https://doi.org/10.1137/23m1550992","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1159-1198, June 2024. <br/> Abstract.In this work we study the splitting distance of a rapidly perturbed pendulum [math] with [math] a [math]-periodic function and [math]. Systems of this kind undergo exponentially small splitting, and, when [math], it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided [math]. Our study focuses on the case [math], and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as [math]. On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency [math] in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with [math] where, for most [math], the perturbation satisfies [math]. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton–Jacobi formalism. The leading exponentially small term appears at order [math], where [math] is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"36 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925923","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 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":"83 1","pages":""},"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":"2011 1","pages":""},"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":"80 1","pages":""},"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":"102 1","pages":""},"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":"49 1","pages":""},"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}
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