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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 668-695, March 2024. Abstract.We present a geometrical description of the symmetries and reduction of the full gravitational 2-body problem after complete averaging over fast angles. Our variables allow for a well-suited formulation in action-angle type coordinates associated with the averaged angles, which provide geometric insight into the problem. After introducing extra fictitious variables and through a symplectic transformation, we move to a singularity-free quaternionic triple-chart. This choice allows for a global chart to avoid the classical singularities associated with angles and renders all the invariants as homogeneous quadratic polynomials. Additionally, it permits one to quickly write the Hamiltonian of the system in terms of the invariants and the Poisson structure at each stage of the reduction process. In contrast with existing literature, the geometrical approach of this research completely describes all the dynamical aspects of the full reduced space since it involves the relative position of the rotational and orbital angular momenta and their orientation, which has yet to be considered in previous studies. Our program includes a preliminary parametric analysis of relative equilibria and a complete description of the fibers in the reconstruction of the reduced system.
{"title":"[math]-Reduction, Relative Equilibria, and Bifurcations for the Full Averaged Model of Two Interacting Rigid Bodies","authors":"F. Crespo, D. E. Espejo, J. C. van der Meer","doi":"10.1137/23m158125x","DOIUrl":"https://doi.org/10.1137/23m158125x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 668-695, March 2024. <br/> Abstract.We present a geometrical description of the symmetries and reduction of the full gravitational 2-body problem after complete averaging over fast angles. Our variables allow for a well-suited formulation in action-angle type coordinates associated with the averaged angles, which provide geometric insight into the problem. After introducing extra fictitious variables and through a symplectic transformation, we move to a singularity-free quaternionic triple-chart. This choice allows for a global chart to avoid the classical singularities associated with angles and renders all the invariants as homogeneous quadratic polynomials. Additionally, it permits one to quickly write the Hamiltonian of the system in terms of the invariants and the Poisson structure at each stage of the reduction process. In contrast with existing literature, the geometrical approach of this research completely describes all the dynamical aspects of the full reduced space since it involves the relative position of the rotational and orbital angular momenta and their orientation, which has yet to be considered in previous studies. Our program includes a preliminary parametric analysis of relative equilibria and a complete description of the fibers in the reconstruction of the reduced system.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139925455","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 641-667, March 2024. Abstract.To model a single-species cognitive movement, we formulate a reaction-diffusion equation with nonlocal spatial memory and investigate its dynamics. We explore the influence of the perceptual scale on the stability and Turing bifurcation. When the random diffusion is dominant, the perceptual scale does not affect the stability, but when the memory-based diffusion is dominant, there exist Turing bifurcations induced by the perceptual scale. Then the joint effect of the perceptual scale and the memory delay on the stability and spatio-temporal dynamics is investigated to show rich spatio-temporal dynamics via Turing–Hopf bifurcation and double Hopf bifurcation. Finally, we apply our analysis to an application and illustrate our theoretical results with numerical simulations.
{"title":"Spatio-temporal Dynamics in a Reaction-Diffusion Equation with Nonlocal Spatial Memory","authors":"Shuyang Xue, Yongli Song, Hao Wang","doi":"10.1137/22m1543860","DOIUrl":"https://doi.org/10.1137/22m1543860","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 641-667, March 2024. <br/>Abstract.To model a single-species cognitive movement, we formulate a reaction-diffusion equation with nonlocal spatial memory and investigate its dynamics. We explore the influence of the perceptual scale on the stability and Turing bifurcation. When the random diffusion is dominant, the perceptual scale does not affect the stability, but when the memory-based diffusion is dominant, there exist Turing bifurcations induced by the perceptual scale. Then the joint effect of the perceptual scale and the memory delay on the stability and spatio-temporal dynamics is investigated to show rich spatio-temporal dynamics via Turing–Hopf bifurcation and double Hopf bifurcation. Finally, we apply our analysis to an application and illustrate our theoretical results with numerical simulations.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139925694","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 696-720, March 2024. Abstract.In this paper, we study a two stream species Lotka–Volterra competition patch model with the patches aligned along a line. The two species are supposed to be identical except for the diffusion rates. For each species, the diffusion rates between patches are the same, while the drift rates vary. Our results show that the convexity of the drift rates has a significant impact on the competition outcomes: if the drift rates are convex, then the species with the larger diffusion rate wins the competition; if the drift rates are concave, then the species with the smaller diffusion rate wins the competition.
{"title":"Evolution of Dispersal in Advective Patchy Environments with Varying Drift Rates","authors":"Shanshan Chen, Jie Liu, Yixiang Wu","doi":"10.1137/22m1542027","DOIUrl":"https://doi.org/10.1137/22m1542027","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 696-720, March 2024. <br/> Abstract.In this paper, we study a two stream species Lotka–Volterra competition patch model with the patches aligned along a line. The two species are supposed to be identical except for the diffusion rates. For each species, the diffusion rates between patches are the same, while the drift rates vary. Our results show that the convexity of the drift rates has a significant impact on the competition outcomes: if the drift rates are convex, then the species with the larger diffusion rate wins the competition; if the drift rates are concave, then the species with the smaller diffusion rate wins the competition.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139925457","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 616-640, March 2024. Abstract. We previously showed that three-dimensional quadratic diffeomorphisms have anti-integrable (AI) limits that correspond to a quadratic correspondence, a pair of one-dimensional maps. At the AI limit the dynamics is conjugate to a full shift on two symbols. Here we consider a more general AI limit, allowing two parameters of the map to go to infinity. We prove the existence of AI states for each symbol sequence for three cases of the quadratic correspondence: parabolas, ellipses, and hyperbolas. A contraction argument gives parameter domains such that this is a bijection, but the correspondence also is observed to apply more generally. We show that orbits of the original map can be obtained by numerical continuation for a volume-contracting case. These results show that periodic AI states evolve into the observed periodic attractors of the diffeomorphism. We also continue a periodic AI state with a symbol sequence chosen so that it continues to an orbit resembling a chaotic attractor that is a 3D version of the classical 2D Hénon attractor.
SIAM 应用动力系统期刊》第 23 卷第 1 期第 616-640 页，2024 年 3 月。 摘要。我们之前证明了三维二次差分变分具有反可积分（AI）极限，它对应于二次对应关系，即一对一维映射。在 AI 极限，动力学共轭于两个符号上的全移。在这里，我们考虑了更一般的 AI 极限，允许映射的两个参数达到无穷大。我们证明了在抛物线、椭圆和双曲线这三种二次对应的情况下，每个符号序列都存在人工智能状态。收缩论证给出了参数域，因此这是一个双射，但也观察到对应关系适用于更广泛的情况。我们证明，在体积收缩的情况下，可以通过数值延续得到原始映射的轨道。这些结果表明，周期性人工智能状态会演化成所观察到的衍射周期性吸引子。我们还用符号序列延续了一个周期性人工智能状态，使其延续到一个类似于混沌吸引子的轨道，而混沌吸引子是经典二维赫农吸引子的三维版本。
{"title":"Connecting Anti-integrability to Attractors for Three-Dimensional Quadratic Diffeomorphisms","authors":"Amanda E. Hampton, James D. Meiss","doi":"10.1137/23m1571897","DOIUrl":"https://doi.org/10.1137/23m1571897","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 616-640, March 2024. <br/> Abstract. We previously showed that three-dimensional quadratic diffeomorphisms have anti-integrable (AI) limits that correspond to a quadratic correspondence, a pair of one-dimensional maps. At the AI limit the dynamics is conjugate to a full shift on two symbols. Here we consider a more general AI limit, allowing two parameters of the map to go to infinity. We prove the existence of AI states for each symbol sequence for three cases of the quadratic correspondence: parabolas, ellipses, and hyperbolas. A contraction argument gives parameter domains such that this is a bijection, but the correspondence also is observed to apply more generally. We show that orbits of the original map can be obtained by numerical continuation for a volume-contracting case. These results show that periodic AI states evolve into the observed periodic attractors of the diffeomorphism. We also continue a periodic AI state with a symbol sequence chosen so that it continues to an orbit resembling a chaotic attractor that is a 3D version of the classical 2D Hénon attractor.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770434","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}
Alan Veliz-Cuba, Vanessa Newsome-Slade, Elena S. Dimitrova
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 592-615, March 2024. Abstract.Due to cost concerns, it is optimal to gain insight into the connectivity of biological and other networks using as few experiments as possible. Data selection for unique network connectivity identification has been an open problem since the introduction of algebraic methods for reverse engineering for almost two decades. In this manuscript we determine what data sets uniquely identify the unsigned wiring diagram corresponding to a system that is discrete in time and space. Furthermore, we answer the question of uniqueness for signed wiring diagrams for Boolean networks. Computationally, unsigned and signed wiring diagrams have been studied separately, and in this manuscript we also show that there exists an ideal capable of encoding both unsigned and signed information. This provides a unified approach to studying reverse engineering that also gives significant computational benefits.
{"title":"A Unified Approach to Reverse Engineering and Data Selection for Unique Network Identification","authors":"Alan Veliz-Cuba, Vanessa Newsome-Slade, Elena S. Dimitrova","doi":"10.1137/22m1540570","DOIUrl":"https://doi.org/10.1137/22m1540570","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 592-615, March 2024. <br/> Abstract.Due to cost concerns, it is optimal to gain insight into the connectivity of biological and other networks using as few experiments as possible. Data selection for unique network connectivity identification has been an open problem since the introduction of algebraic methods for reverse engineering for almost two decades. In this manuscript we determine what data sets uniquely identify the unsigned wiring diagram corresponding to a system that is discrete in time and space. Furthermore, we answer the question of uniqueness for signed wiring diagrams for Boolean networks. Computationally, unsigned and signed wiring diagrams have been studied separately, and in this manuscript we also show that there exists an ideal capable of encoding both unsigned and signed information. This provides a unified approach to studying reverse engineering that also gives significant computational benefits.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770627","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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