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":"29 1","pages":""},"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":"21 1","pages":""},"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":"102 1","pages":""},"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":"25 1","pages":""},"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":"4 1","pages":""},"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}
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 813-854, March 2024. Abstract.For polynomial ODE models, we introduce and discuss the concepts of exact and approximate conservation laws, which are the first integrals of the full and truncated sets of ODEs. For fast-slow systems, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the state space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states. Complete sets of conservation laws can be used for model order reduction and for studying the multistationarity of the model. We provide algorithmic methods for computing linear, monomial, and polynomial conservation laws of polynomial ODE models and for testing their completeness. The resulting conservation laws and their completeness are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gröbner systems and syzygies.
{"title":"A Computational Approach to Polynomial Conservation Laws","authors":"Aurélien Desoeuvres, Alexandru Iosif, Christoph Lüders, Ovidiu Radulescu, Hamid Rahkooy, Matthias Seiß, Thomas Sturm","doi":"10.1137/22m1544014","DOIUrl":"https://doi.org/10.1137/22m1544014","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 813-854, March 2024. <br/> Abstract.For polynomial ODE models, we introduce and discuss the concepts of exact and approximate conservation laws, which are the first integrals of the full and truncated sets of ODEs. For fast-slow systems, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the state space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states. Complete sets of conservation laws can be used for model order reduction and for studying the multistationarity of the model. We provide algorithmic methods for computing linear, monomial, and polynomial conservation laws of polynomial ODE models and for testing their completeness. The resulting conservation laws and their completeness are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gröbner systems and syzygies.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140116744","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 791-812, March 2024. Abstract.Moment systems arise in a wide range of contexts and applications, e.g., in network modeling of complex systems. Since moment systems consist of a high or even infinite number of coupled equations, an indispensable step in obtaining a low-dimensional representation that is amenable to further analysis is, in many cases, to select a moment closure. A moment closure consists of a set of approximations that express certain higher-order moments in terms of lower-order ones, so that applying those leads to a closed system of equations for only the lower-order moments. Closures are frequently found drawing on intuition and heuristics to come up with quantitatively good approximations. In contrast to that, we propose an alternative approach where we instead focus on closures giving rise to certain qualitative features, such as bifurcations. Importantly, this fundamental change of perspective provides one with the possibility of classifying moment closures rigorously in regard to these features. This makes the design and selection of closures more algorithmic, precise, and reliable. In this work, we carefully study the moment systems that arise in the mean-field descriptions of two widely known network dynamical systems, the SIS epidemic and the adaptive voter model. We derive conditions that any moment closure has to satisfy so that the corresponding closed systems exhibit the transcritical bifurcation that one expects in these systems coming from the stochastic particle model.
{"title":"Preserving Bifurcations through Moment Closures","authors":"Christian Kuehn, Jan Mölter","doi":"10.1137/23m158440x","DOIUrl":"https://doi.org/10.1137/23m158440x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 791-812, March 2024. <br/> Abstract.Moment systems arise in a wide range of contexts and applications, e.g., in network modeling of complex systems. Since moment systems consist of a high or even infinite number of coupled equations, an indispensable step in obtaining a low-dimensional representation that is amenable to further analysis is, in many cases, to select a moment closure. A moment closure consists of a set of approximations that express certain higher-order moments in terms of lower-order ones, so that applying those leads to a closed system of equations for only the lower-order moments. Closures are frequently found drawing on intuition and heuristics to come up with quantitatively good approximations. In contrast to that, we propose an alternative approach where we instead focus on closures giving rise to certain qualitative features, such as bifurcations. Importantly, this fundamental change of perspective provides one with the possibility of classifying moment closures rigorously in regard to these features. This makes the design and selection of closures more algorithmic, precise, and reliable. In this work, we carefully study the moment systems that arise in the mean-field descriptions of two widely known network dynamical systems, the SIS epidemic and the adaptive voter model. We derive conditions that any moment closure has to satisfy so that the corresponding closed systems exhibit the transcritical bifurcation that one expects in these systems coming from the stochastic particle model.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"67 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140116732","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}
Pedro Abdalla, Afonso S. Bandeira, Clara Invernizzi
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 779-790, March 2024. Abstract. The Kuramoto model is a classical mathematical model in the field of nonlinear dynamical systems that describes the evolution of coupled oscillators in a network that may reach a synchronous state. The relationship between the network’s topology and whether the oscillators synchronize is a central question in the field of synchronization, and random graphs are often employed as a proxy for complex networks. On the other hand, the random graphs on which the Kuramoto model is rigorously analyzed in the literature are homogeneous models and fail to capture the underlying geometric structure that appears in several examples. In this work, we leverage tools from random matrix theory, random graphs, and mathematical statistics to prove that the Kuramoto model on a random geometric graph on the sphere synchronizes with probability tending to one as the number of nodes tends to infinity. To the best of our knowledge, this is the first rigorous result for the Kuramoto model on random geometric graphs.
{"title":"Guarantees for Spontaneous Synchronization on Random Geometric Graphs","authors":"Pedro Abdalla, Afonso S. Bandeira, Clara Invernizzi","doi":"10.1137/23m1559270","DOIUrl":"https://doi.org/10.1137/23m1559270","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 779-790, March 2024. <br/> Abstract. The Kuramoto model is a classical mathematical model in the field of nonlinear dynamical systems that describes the evolution of coupled oscillators in a network that may reach a synchronous state. The relationship between the network’s topology and whether the oscillators synchronize is a central question in the field of synchronization, and random graphs are often employed as a proxy for complex networks. On the other hand, the random graphs on which the Kuramoto model is rigorously analyzed in the literature are homogeneous models and fail to capture the underlying geometric structure that appears in several examples. In this work, we leverage tools from random matrix theory, random graphs, and mathematical statistics to prove that the Kuramoto model on a random geometric graph on the sphere synchronizes with probability tending to one as the number of nodes tends to infinity. To the best of our knowledge, this is the first rigorous result for the Kuramoto model on random geometric graphs.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"81 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054476","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}
Haotian Wang, Hujiang Yang, Xiankui Meng, Ye Tian, Wenjun Liu
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 748-778, March 2024. Abstract. This paper studies the soliton dynamics for the Rabi-coupled Gross–Pitaevskii model in multicomponent Bose–Einstein condensates. The model has variable nonlinearities and external potentials and is used to construct a complex multisoliton in an explicit form. The variable nonlinearity and external potential cause the soliton to compress and change its velocity, respectively. A new generalized similarity transformation is proposed to eliminate the [math] Rabi-coupled terms in the [math]-component model, which can make the Hirota bilinear method be applied to obtain multisoliton solutions. The bound state of the two-soliton forms the soliton molecule under velocity resonance. Asymptotic analysis can give the asymptotic expressions of each single soliton in multisoliton solutions, which can clearly give each soliton’s width, velocity, amplitude, and energy; these parameters can control multisolitons. When the solitons’ relative velocity or the solitons’ width is large, the interferogram between solitons will be observed. Numerical simulation shows that these solitons can steadily propagate. It is easy for the soliton molecule and interference dynamics to occur because of the controlled soliton. Since the coupled Gross–Pitaevskii equation describes the mechanics of matter waves in Bose–Einstein condensates, it is proved that we can observe the stable solitons and soliton molecules in Bose–Einstein condensates. The method and results presented in this paper are also common to other similar models. When observing particle multiple distributions, quantum interferometry, and interferometers, the results presented and the model in this paper can provide a reference for these applications.
{"title":"Dynamics of Controllable Matter-Wave Solitons and Soliton Molecules for a Rabi-Coupled Gross–Pitaevskii Equation with Temporally and Spatially Modulated Coefficients","authors":"Haotian Wang, Hujiang Yang, Xiankui Meng, Ye Tian, Wenjun Liu","doi":"10.1137/23m155551x","DOIUrl":"https://doi.org/10.1137/23m155551x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 748-778, March 2024. <br/> Abstract. This paper studies the soliton dynamics for the Rabi-coupled Gross–Pitaevskii model in multicomponent Bose–Einstein condensates. The model has variable nonlinearities and external potentials and is used to construct a complex multisoliton in an explicit form. The variable nonlinearity and external potential cause the soliton to compress and change its velocity, respectively. A new generalized similarity transformation is proposed to eliminate the [math] Rabi-coupled terms in the [math]-component model, which can make the Hirota bilinear method be applied to obtain multisoliton solutions. The bound state of the two-soliton forms the soliton molecule under velocity resonance. Asymptotic analysis can give the asymptotic expressions of each single soliton in multisoliton solutions, which can clearly give each soliton’s width, velocity, amplitude, and energy; these parameters can control multisolitons. When the solitons’ relative velocity or the solitons’ width is large, the interferogram between solitons will be observed. Numerical simulation shows that these solitons can steadily propagate. It is easy for the soliton molecule and interference dynamics to occur because of the controlled soliton. Since the coupled Gross–Pitaevskii equation describes the mechanics of matter waves in Bose–Einstein condensates, it is proved that we can observe the stable solitons and soliton molecules in Bose–Einstein condensates. The method and results presented in this paper are also common to other similar models. When observing particle multiple distributions, quantum interferometry, and interferometers, the results presented and the model in this paper can provide a reference for these applications.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"31 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140011084","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 721-747, March 2024. Abstract.A class of four-component reaction-diffusion systems are studied in one spatial dimension, with one of four specific reaction kinetics. Models of this type seek to capture the interaction between active and inactive forms of two G-proteins, known as ROPs in plants, thought to underly cellular polarity formation. The systems conserve total concentration of each ROP, which enables reduction to simple canonical forms when one seeks conditions for homogeneous equilibria or heteroclinic connections between them. Transitions between different multiplicities of such states are classified using a novel application of catastrophe theory. For the time-dependent problem, the heteroclinic connections represent so-called wave-pinned states that separate regions of the domain with different ROP concentrations. It is shown numerically how the form of wave-pinning reached can be predicted as a function of the domain size and initial total ROP concentrations. This leads to state diagrams of different polarity forms as a function of total concentrations and system parameters.
{"title":"Wave-Pinned Patterns for Cell Polarity—A Catastrophe Theory Explanation","authors":"Fahad Al Saadi, Alan Champneys, Mike R. Jeffrey","doi":"10.1137/22m1509758","DOIUrl":"https://doi.org/10.1137/22m1509758","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 721-747, March 2024. <br/> Abstract.A class of four-component reaction-diffusion systems are studied in one spatial dimension, with one of four specific reaction kinetics. Models of this type seek to capture the interaction between active and inactive forms of two G-proteins, known as ROPs in plants, thought to underly cellular polarity formation. The systems conserve total concentration of each ROP, which enables reduction to simple canonical forms when one seeks conditions for homogeneous equilibria or heteroclinic connections between them. Transitions between different multiplicities of such states are classified using a novel application of catastrophe theory. For the time-dependent problem, the heteroclinic connections represent so-called wave-pinned states that separate regions of the domain with different ROP concentrations. It is shown numerically how the form of wave-pinning reached can be predicted as a function of the domain size and initial total ROP concentrations. This leads to state diagrams of different polarity forms as a function of total concentrations and system parameters.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"44 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979658","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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