SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 553-591, March 2024. Abstract. In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov–Takens bifurcation in classical delay differential equations. Using an approximation to the homoclinic solutions derived with a generalized Lindstedt–Poincaré method, we develop a method to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas, which have been implemented in the freely available bifurcation software package DDE-BifTool. The effectiveness is demonstrated on various models
{"title":"Bifurcation Analysis of Bogdanov–Takens Bifurcations in Delay Differential Equations","authors":"M. M. Bosschaert, Yu. A. Kuznetsov","doi":"10.1137/22m1527532","DOIUrl":"https://doi.org/10.1137/22m1527532","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 553-591, March 2024. <br/> Abstract. In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov–Takens bifurcation in classical delay differential equations. Using an approximation to the homoclinic solutions derived with a generalized Lindstedt–Poincaré method, we develop a method to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas, which have been implemented in the freely available bifurcation software package DDE-BifTool. The effectiveness is demonstrated on various models","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646702","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 470-504, March 2024. Abstract. We propose a general strategy for reduced order modeling of systems that display highly nonlinear oscillations. By considering a continuous family of forced periodic orbits defined in relation to a stable fixed point and subsequently leveraging phase-amplitude-based reduction strategies, we arrive at a low order model capable of accurately capturing nonlinear oscillations resulting from arbitrary external inputs. In the limit that oscillations are small, the system dynamics relax to those obtained from local linearization, i.e., that can be fully described using linear eigenmodes. For larger amplitude oscillations, the behavior can be understood in terms of the dynamics of a small number of nonlinear modes. We illustrate the proposed strategy in a variety of examples yielding results that are substantially better than those obtained using standard linearization-based techniques.
{"title":"Reduced Order Characterization of Nonlinear Oscillations Using an Adaptive Phase-Amplitude Coordinate Framework","authors":"Dan Wilson, Kai Sun","doi":"10.1137/23m1551699","DOIUrl":"https://doi.org/10.1137/23m1551699","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 470-504, March 2024. <br/> Abstract. We propose a general strategy for reduced order modeling of systems that display highly nonlinear oscillations. By considering a continuous family of forced periodic orbits defined in relation to a stable fixed point and subsequently leveraging phase-amplitude-based reduction strategies, we arrive at a low order model capable of accurately capturing nonlinear oscillations resulting from arbitrary external inputs. In the limit that oscillations are small, the system dynamics relax to those obtained from local linearization, i.e., that can be fully described using linear eigenmodes. For larger amplitude oscillations, the behavior can be understood in terms of the dynamics of a small number of nonlinear modes. We illustrate the proposed strategy in a variety of examples yielding results that are substantially better than those obtained using standard linearization-based techniques.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139583644","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 505-552, March 2024. Abstract. Traditional sensitivity analysis methods fail for chaotic systems due to the unstable characteristics of the linearized equations. To overcome these issues two methods have been developed in the literature, one being the shadowing approach, which results in a minimization problem, and the other being numerical viscosity, where a damping term is added to the linearized equations to suppress the instability. The shadowing approach is computationally expensive but produces accurate sensitivities, while numerical viscosity can produce less accurate sensitivities but with significantly reduced computational cost. However, it is not fully clear how the solutions generated by these two approaches compare to each other. In this work we aim to bridge this gap by introducing a control term, found with optimal control theory techniques, to prevent the exponential growth of solution of the linearized equations. We will refer to this method as optimal control shadowing. We investigate the computational aspects and performance of this new method on the Lorenz and Kuramoto–Sivashinsky systems and compare its performance with simple numerical viscosity schemes. We show that the tangent solution generated by the proposed approach is similar to that generated by shadowing methods, suggesting that optimal control attempts to stabilize the unstable shadowing direction. Further, for the spatially extended system, we examine the energy budget of the tangent equation and show that the control term found via the solution of the optimal control problem acts only at length scales where production of tangent energy dominates dissipation, which is not necessarily the case for the numerical viscosity methods.
{"title":"On the Application of Optimal Control Techniques to the Shadowing Approach for Time Averaged Sensitivity Analysis of Chaotic Systems","authors":"Rhys E. Gilbert, Davide Lasagna","doi":"10.1137/23m1550219","DOIUrl":"https://doi.org/10.1137/23m1550219","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 505-552, March 2024. <br/> Abstract. Traditional sensitivity analysis methods fail for chaotic systems due to the unstable characteristics of the linearized equations. To overcome these issues two methods have been developed in the literature, one being the shadowing approach, which results in a minimization problem, and the other being numerical viscosity, where a damping term is added to the linearized equations to suppress the instability. The shadowing approach is computationally expensive but produces accurate sensitivities, while numerical viscosity can produce less accurate sensitivities but with significantly reduced computational cost. However, it is not fully clear how the solutions generated by these two approaches compare to each other. In this work we aim to bridge this gap by introducing a control term, found with optimal control theory techniques, to prevent the exponential growth of solution of the linearized equations. We will refer to this method as optimal control shadowing. We investigate the computational aspects and performance of this new method on the Lorenz and Kuramoto–Sivashinsky systems and compare its performance with simple numerical viscosity schemes. We show that the tangent solution generated by the proposed approach is similar to that generated by shadowing methods, suggesting that optimal control attempts to stabilize the unstable shadowing direction. Further, for the spatially extended system, we examine the energy budget of the tangent equation and show that the control term found via the solution of the optimal control problem acts only at length scales where production of tangent energy dominates dissipation, which is not necessarily the case for the numerical viscosity methods.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139583339","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 410-439, March 2024. Abstract.This paper provides for the first time correct third-order homoclinic predictors in [math]-dimensional ODEs near a generic Bogdanov–Takens bifurcation point, which can be used to start the numerical continuation of the appearing homoclinic orbits. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt–Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is derived rigorously. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt–Poincaré) to approximate the homoclinic solution near Bogdanov–Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.
{"title":"Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov–Takens Bifurcation","authors":"Maikel M. Bosschaert, Yuri A. Kuznetsov","doi":"10.1137/22m151354x","DOIUrl":"https://doi.org/10.1137/22m151354x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 410-439, March 2024. <br/> Abstract.This paper provides for the first time correct third-order homoclinic predictors in [math]-dimensional ODEs near a generic Bogdanov–Takens bifurcation point, which can be used to start the numerical continuation of the appearing homoclinic orbits. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt–Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is derived rigorously. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt–Poincaré) to approximate the homoclinic solution near Bogdanov–Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554944","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}
Megan R. Ebers, Katherine M. Steele, J. Nathan Kutz
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 440-469, March 2024. Abstract.Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations often result in discrepancies between the model and sensor-based measurements of the system, revealing the approximate nature of the equations and/or the signal-to-noise ratio of the sensor itself. In modern dynamical systems, such discrepancies between model and measurement can lead to poor quantification, often undermining the ability to produce accurate and precise control algorithms. We introduce a discrepancy modeling framework to identify the missing physics and resolve the model-measurement mismatch with two distinct approaches: (i) by learning a model for the evolution of systematic state-space residual, and (ii) by discovering a model for the deterministic dynamical error. Regardless of approach, a common suite of data-driven model discovery methods can be used. Specifically, we use four fundamentally different methods to demonstrate the mathematical implementations of discrepancy modeling: (i) the sparse identification of nonlinear dynamics, (ii) dynamic mode decomposition, (iii) Gaussian process regression, and (iv) neural networks. The choice of method depends on one’s intent (e.g., mechanistic interpretability) for discrepancy modeling, sensor measurement characteristics (e.g., quantity, quality, resolution), and constraints imposed by practical applications (e.g., state- or dynamical-space operability). We demonstrate the utility and suitability for discrepancy modeling using the suite of data-driven modeling methods on four dynamical systems under varying signal-to-noise ratios. Finally, we emphasize structural shortcomings of each discrepancy modeling approach depending on error type. In summary, if the true dynamics are unknown (i.e., an imperfect model), one should learn a discrepancy model of the missing physics in the dynamical space. Yet, if the true dynamics are known yet model-measurement mismatch still exists, one should learn a discrepancy model in the state space.
{"title":"Discrepancy Modeling Framework: Learning Missing Physics, Modeling Systematic Residuals, and Disambiguating between Deterministic and Random Effects","authors":"Megan R. Ebers, Katherine M. Steele, J. Nathan Kutz","doi":"10.1137/22m148375x","DOIUrl":"https://doi.org/10.1137/22m148375x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 440-469, March 2024. <br/> Abstract.Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations often result in discrepancies between the model and sensor-based measurements of the system, revealing the approximate nature of the equations and/or the signal-to-noise ratio of the sensor itself. In modern dynamical systems, such discrepancies between model and measurement can lead to poor quantification, often undermining the ability to produce accurate and precise control algorithms. We introduce a discrepancy modeling framework to identify the missing physics and resolve the model-measurement mismatch with two distinct approaches: (i) by learning a model for the evolution of systematic state-space residual, and (ii) by discovering a model for the deterministic dynamical error. Regardless of approach, a common suite of data-driven model discovery methods can be used. Specifically, we use four fundamentally different methods to demonstrate the mathematical implementations of discrepancy modeling: (i) the sparse identification of nonlinear dynamics, (ii) dynamic mode decomposition, (iii) Gaussian process regression, and (iv) neural networks. The choice of method depends on one’s intent (e.g., mechanistic interpretability) for discrepancy modeling, sensor measurement characteristics (e.g., quantity, quality, resolution), and constraints imposed by practical applications (e.g., state- or dynamical-space operability). We demonstrate the utility and suitability for discrepancy modeling using the suite of data-driven modeling methods on four dynamical systems under varying signal-to-noise ratios. Finally, we emphasize structural shortcomings of each discrepancy modeling approach depending on error type. In summary, if the true dynamics are unknown (i.e., an imperfect model), one should learn a discrepancy model of the missing physics in the dynamical space. Yet, if the true dynamics are known yet model-measurement mismatch still exists, one should learn a discrepancy model in the state space.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139555271","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}
Bogdan Batko, Marcio Gameiro, Ying Hung, William Kalies, Konstantin Mischaikow, Ewerton Vieira
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 383-409, March 2024. Abstract.We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system (assumed to be a sample path of the GP). The focus of this paper is on explaining the ideas, thus we restrict our examples to one-dimensional systems and show how to capture the existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics.
{"title":"Identifying Nonlinear Dynamics with High Confidence from Sparse Data","authors":"Bogdan Batko, Marcio Gameiro, Ying Hung, William Kalies, Konstantin Mischaikow, Ewerton Vieira","doi":"10.1137/23m1560252","DOIUrl":"https://doi.org/10.1137/23m1560252","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 383-409, March 2024. <br/> Abstract.We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system (assumed to be a sample path of the GP). The focus of this paper is on explaining the ideas, thus we restrict our examples to one-dimensional systems and show how to capture the existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554919","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 358-382, March 2024. Abstract. This paper is mainly concerned with limiting behaviors of invariant measures for neural field lattice models in a random environment. First of all, we consider the convergence relation of invariant measures between the stochastic neural field lattice model and the corresponding deterministic model in weighted spaces, and prove any limit of a sequence of invariant measures of such a lattice model must be an invariant measure of its limiting system as the noise intensity tends to zero. Then we are devoted to studying the numerical approximation of invariant measures of such a stochastic neural lattice model. To this end, we first consider convergence of invariant measures between such a neural lattice model and the system with neurons only interacting with its n-neighborhood; then we further prove the convergence relation of invariant measures between the system with an n-neighborhood and its finite dimensional truncated system. By this procedure, the invariant measure of the stochastic neural lattice models can be approximated by the numerical invariant measure of a finite dimensional truncated system based on the backward Euler–Maruyama (BEM) scheme. Therefore, the invariant measure of a deterministic neural field lattice model can be observed by the invariant measure of the BEM scheme when the noise is not negligible.
SIAM 应用动力系统期刊》，第 23 卷第 1 期，第 358-382 页，2024 年 3 月。 摘要本文主要研究随机环境下神经场晶格模型不变度量的极限行为。首先，我们考虑了随机神经场网格模型与相应的确定性模型在加权空间中的不变量度量收敛关系，并证明了当噪声强度趋于零时，该网格模型不变量度量序列的任何极限必定是其极限系统的不变量度量。然后，我们将致力于研究这种随机神经网格模型不变量的数值逼近。为此，我们首先考虑这种神经网格模型与神经元只与其 n 邻域相互作用的系统之间的不变量度量的收敛性；然后，我们进一步证明具有 n 邻域的系统与其有限维截断系统之间的不变量度量的收敛关系。通过这一过程，随机神经网格模型的不变度量可以用基于后向欧拉-马鲁山（BEM）方案的有限维截断系统的数值不变度量来近似。因此，当噪声不可忽略时，确定性神经场网格模型的不变度量可以通过 BEM 方案的不变度量来观察。
{"title":"Convergence and Approximation of Invariant Measures for Neural Field Lattice Models under Noise Perturbation","authors":"Tomas Caraballo, Zhang Chen, Lingyu Li","doi":"10.1137/23m157137x","DOIUrl":"https://doi.org/10.1137/23m157137x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 358-382, March 2024. <br/> Abstract. This paper is mainly concerned with limiting behaviors of invariant measures for neural field lattice models in a random environment. First of all, we consider the convergence relation of invariant measures between the stochastic neural field lattice model and the corresponding deterministic model in weighted spaces, and prove any limit of a sequence of invariant measures of such a lattice model must be an invariant measure of its limiting system as the noise intensity tends to zero. Then we are devoted to studying the numerical approximation of invariant measures of such a stochastic neural lattice model. To this end, we first consider convergence of invariant measures between such a neural lattice model and the system with neurons only interacting with its n-neighborhood; then we further prove the convergence relation of invariant measures between the system with an n-neighborhood and its finite dimensional truncated system. By this procedure, the invariant measure of the stochastic neural lattice models can be approximated by the numerical invariant measure of a finite dimensional truncated system based on the backward Euler–Maruyama (BEM) scheme. Therefore, the invariant measure of a deterministic neural field lattice model can be observed by the invariant measure of the BEM scheme when the noise is not negligible.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554595","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}
Olivia Cannon, Ty Bondurant, Malindi Whyte, Arnd Scheel
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 297-324, March 2024. Abstract. We investigate the effect of bias on the formation and dynamics of opinion clusters in the bounded confidence model. For weak bias, we quantify the change in average opinion and potential dispersion and decrease in cluster size. For nonlinear bias modeling self-incitement, we establish coherent drifting motion of clusters on a background of uniform opinion distribution for biases below a critical threshold where clusters dissolve. Technically, we use geometric singular perturbation theory to derive drift speeds, we rely on a nonlocal center manifold analysis to construct drifting clusters near threshold, and we implement numerical continuation in a forward-backward delay equation to connect asymptotic regimes.
{"title":"Shifting Consensus in a Biased Compromise Model","authors":"Olivia Cannon, Ty Bondurant, Malindi Whyte, Arnd Scheel","doi":"10.1137/23m1552346","DOIUrl":"https://doi.org/10.1137/23m1552346","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 297-324, March 2024. <br/> Abstract. We investigate the effect of bias on the formation and dynamics of opinion clusters in the bounded confidence model. For weak bias, we quantify the change in average opinion and potential dispersion and decrease in cluster size. For nonlinear bias modeling self-incitement, we establish coherent drifting motion of clusters on a background of uniform opinion distribution for biases below a critical threshold where clusters dissolve. Technically, we use geometric singular perturbation theory to derive drift speeds, we rely on a nonlocal center manifold analysis to construct drifting clusters near threshold, and we implement numerical continuation in a forward-backward delay equation to connect asymptotic regimes.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554599","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 325-357, March 2024. Abstract. Generalized mass-action systems are power-law dynamical systems arising from chemical reaction networks. Essentially, every nonnegative ODE model used in chemistry and biology (for example, in ecology and epidemiology) and even in economics and engineering can be written in this form. Previous results have focused on existence and uniqueness of special steady states (complex-balanced equilibria) for all rate constants, thereby ruling out multiple (special) steady states. Recently, necessary conditions for linear stability have been obtained. In this work, we provide sufficient conditions for the linear stability of complex-balanced equilibria for all rate constants (and also for the nonexistence of other steady states). In particular, via sign vector conditions (on the stoichiometric coefficients and kinetic orders), we guarantee that the Jacobian matrix is a [math]-matrix. Technically, we use a new decomposition of the graph Laplacian which allows us to consider orders of (generalized) monomials. Alternatively, we use cycle decomposition which allows a linear parametrization of all Jacobian matrices. In any case, we guarantee stability without explicit computation of steady states. We illustrate our results in examples from chemistry and biology: generalized Lotka–Volterra systems and SIR models, a two-component signaling system, and an enzymatic futile cycle.
{"title":"Sufficient Conditions for Linear Stability of Complex-Balanced Equilibria in Generalized Mass-Action Systems","authors":"Stefan Müller, Georg Regensburger","doi":"10.1137/22m154260x","DOIUrl":"https://doi.org/10.1137/22m154260x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 325-357, March 2024. <br/> Abstract. Generalized mass-action systems are power-law dynamical systems arising from chemical reaction networks. Essentially, every nonnegative ODE model used in chemistry and biology (for example, in ecology and epidemiology) and even in economics and engineering can be written in this form. Previous results have focused on existence and uniqueness of special steady states (complex-balanced equilibria) for all rate constants, thereby ruling out multiple (special) steady states. Recently, necessary conditions for linear stability have been obtained. In this work, we provide sufficient conditions for the linear stability of complex-balanced equilibria for all rate constants (and also for the nonexistence of other steady states). In particular, via sign vector conditions (on the stoichiometric coefficients and kinetic orders), we guarantee that the Jacobian matrix is a [math]-matrix. Technically, we use a new decomposition of the graph Laplacian which allows us to consider orders of (generalized) monomials. Alternatively, we use cycle decomposition which allows a linear parametrization of all Jacobian matrices. In any case, we guarantee stability without explicit computation of steady states. We illustrate our results in examples from chemistry and biology: generalized Lotka–Volterra systems and SIR models, a two-component signaling system, and an enzymatic futile cycle.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554630","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}
Aurélien Desoeuvres, Alexandru Iosif, Christoph Lüders, Ovidiu Radulescu, Hamid Rahkooy, Matthias Seiß, Thomas Sturm
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 256-296, March 2024. Abstract. Model reduction of fast-slow chemical reaction networks based on the quasi-steady state approximation fails when the fast subsystem has first integrals. We call these first integrals approximate conservation laws. In order to define fast subsystems and identify approximate conservation laws, we use ideas from tropical geometry. We prove that any approximate conservation law evolves more slowly than all the species involved in it and therefore represents a supplementary slow variable in an extended system. By elimination of some variables of the extended system, we obtain networks without approximate conservation laws, which can be reduced by standard singular perturbation methods. The field of applications of approximate conservation laws covers the quasi-equilibrium approximation, which is well known in biochemistry. We discuss reductions of slow-fast as well as multiple timescale systems. Networks with multiple timescales have hierarchical relaxation. At a given timescale, our multiple timescale reduction method defines three subsystems composed of (i) slaved fast variables satisfying algebraic equations, (ii) slow driving variables satisfying reduced ordinary differential equations, and (iii) quenched much slower variables that are constant. The algebraic equations satisfied by fast variables define chains of nested normally hyperbolic invariant manifolds. In such chains, faster manifolds are of higher dimension and contain the slower manifolds. Our reduction methods are introduced algorithmically for networks with monomial reaction rates and linear, monomial, or polynomial approximate conservation laws. We propose symbolic algorithms to reshape and rescale the networks such that geometric singular perturbation theory can be applied to them, test the applicability of the theory, and finally reduce the networks. As a proof of concept, we apply this method to a model of the TGF-beta signaling pathway.
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