Tamal K. Dey, Michał Lipiński, Marian Mrozek, Ryan Slechta
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 81-97, March 2024. Abstract. Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the context of a rapidly growing data science that requires new mathematical tools for discrete data. Toward this goal, the classical theory for connection matrices has been adapted to combinatorial frameworks that facilitate computation. We develop an efficient persistence-like algorithm to compute a connection matrix from a given combinatorial (multi) vector field on a simplicial complex. This algorithm requires a single pass, improving upon a known algorithm that runs an implicit recursion executing two passes at each level. Overall, the new algorithm is more simple, direct, and efficient than the state-of-the-art. Because of the algorithm’s similarity to the persistence algorithm, one may take advantage of various software optimizations from topological data analysis.
{"title":"Computing Connection Matrices via Persistence-Like Reductions","authors":"Tamal K. Dey, Michał Lipiński, Marian Mrozek, Ryan Slechta","doi":"10.1137/23m1562469","DOIUrl":"https://doi.org/10.1137/23m1562469","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 81-97, March 2024. <br/> Abstract. Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the context of a rapidly growing data science that requires new mathematical tools for discrete data. Toward this goal, the classical theory for connection matrices has been adapted to combinatorial frameworks that facilitate computation. We develop an efficient persistence-like algorithm to compute a connection matrix from a given combinatorial (multi) vector field on a simplicial complex. This algorithm requires a single pass, improving upon a known algorithm that runs an implicit recursion executing two passes at each level. Overall, the new algorithm is more simple, direct, and efficient than the state-of-the-art. Because of the algorithm’s similarity to the persistence algorithm, one may take advantage of various software optimizations from topological data analysis.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"7 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102951","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 50-80, March 2024. Abstract. Two competing types of interactions often play an important part in shaping system behavior, such as activatory and inhibitory functions in biological systems. Hence, signed networks, where each connection can be either positive or negative, have become popular models over recent years. However, the primary focus of the literature is on the unweighted and structurally balanced ones, where all cycles have an even number of negative edges. Hence here, we first introduce a classification of signed networks into balanced, antibalanced, or strictly unbalanced ones, and then characterize each type of signed networks in terms of the spectral properties of the signed weighted adjacency matrix. In particular, we show that the spectral radius of the matrix with signs is smaller than that without if and only if the signed network is strictly unbalanced. These properties are important to understand the dynamics on signed networks, both linear and nonlinear ones. Specifically, we find consistent patterns in a linear and a nonlinear dynamics theoretically, depending on their type of balance. We also propose two measures to further characterize strictly unbalanced networks, motivated by perturbation theory. Finally, we numerically verify these properties through experiments on both synthetic and real networks.
{"title":"Spreading and Structural Balance on Signed Networks","authors":"Yu Tian, Renaud Lambiotte","doi":"10.1137/22m1542325","DOIUrl":"https://doi.org/10.1137/22m1542325","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 50-80, March 2024. <br/> Abstract. Two competing types of interactions often play an important part in shaping system behavior, such as activatory and inhibitory functions in biological systems. Hence, signed networks, where each connection can be either positive or negative, have become popular models over recent years. However, the primary focus of the literature is on the unweighted and structurally balanced ones, where all cycles have an even number of negative edges. Hence here, we first introduce a classification of signed networks into balanced, antibalanced, or strictly unbalanced ones, and then characterize each type of signed networks in terms of the spectral properties of the signed weighted adjacency matrix. In particular, we show that the spectral radius of the matrix with signs is smaller than that without if and only if the signed network is strictly unbalanced. These properties are important to understand the dynamics on signed networks, both linear and nonlinear ones. Specifically, we find consistent patterns in a linear and a nonlinear dynamics theoretically, depending on their type of balance. We also propose two measures to further characterize strictly unbalanced networks, motivated by perturbation theory. Finally, we numerically verify these properties through experiments on both synthetic and real networks.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"23 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102950","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}
L. I. Allen, T. G. Molnár, Z. Dombóvári, S. J. Hogan
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 1-25, March 2024. Abstract. In this paper, we analyze the celebrated Haken–Kelso–Bunz model, describing the dynamics of bimanual coordination, in the presence of delay. We study the linear dynamics, stability, nonlinear behavior, and bifurcations of this model by both theoretical and numerical analysis. We calculate in-phase and antiphase limit cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis and center manifold reduction. Moreover, we uncover further details on the global dynamic behavior by numerical continuation, including the occurrence of limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions.
{"title":"The Effects of Delay on the HKB Model of Human Motor Coordination","authors":"L. I. Allen, T. G. Molnár, Z. Dombóvári, S. J. Hogan","doi":"10.1137/22m1531518","DOIUrl":"https://doi.org/10.1137/22m1531518","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 1-25, March 2024. <br/> Abstract. In this paper, we analyze the celebrated Haken–Kelso–Bunz model, describing the dynamics of bimanual coordination, in the presence of delay. We study the linear dynamics, stability, nonlinear behavior, and bifurcations of this model by both theoretical and numerical analysis. We calculate in-phase and antiphase limit cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis and center manifold reduction. Moreover, we uncover further details on the global dynamic behavior by numerical continuation, including the occurrence of limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139093876","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 26-49, March 2024. Abstract. Contact defects are time-periodic patterns in one space dimension that resemble spatially homogeneous oscillations with a defect embedded in their core region. For theoretical and numerical purposes, it is important to understand whether these defects persist when the domain is truncated to large spatial intervals, supplemented by appropriate boundary conditions. The present work shows that truncated contact defects exist and are unique on sufficiently large spatial intervals.
{"title":"Truncation of Contact Defects in Reaction-Diffusion Systems","authors":"Milen Ivanov, Björn Sandstede","doi":"10.1137/23m1546257","DOIUrl":"https://doi.org/10.1137/23m1546257","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 26-49, March 2024. <br/> Abstract. Contact defects are time-periodic patterns in one space dimension that resemble spatially homogeneous oscillations with a defect embedded in their core region. For theoretical and numerical purposes, it is important to understand whether these defects persist when the domain is truncated to large spatial intervals, supplemented by appropriate boundary conditions. The present work shows that truncated contact defects exist and are unique on sufficiently large spatial intervals.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"22 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139093581","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 22, Issue 4, Page 3390-3430, December 2023. Abstract. For many engineered systems it is important to assess vulnerability to potential disturbances in order to ensure reliable operation. Whether the system will recover from a particular finite-time disturbance to a desired stable equilibrium point depends on uncertain and time-varying system parameter values. Therefore, it is valuable to determine, for specific fixed disturbances, the margins for safe operation: the smallest change in parameter values that would cause the system to become vulnerable to the disturbance. The natural setting for this problem is a parameter-dependent vector field with a family of stable equilibria and a parameter-dependent initial condition representing the disturbance. The system recovers for a particular parameter value if its initial condition lies within the region of attraction of the desired stable equilibrium point. Prior work has developed algorithms for numerically computing the margins for safe operation. However, the theoretical guarantees provided for these methods require a very restrictive assumption: that the nonwandering set in the region of attraction boundary is stable under perturbations to the vector field. This assumption is generally intractable to verify, so feasibility of the above algorithms cannot be determined in advance, and even when these algorithms do converge their convergence to the correct values cannot be guaranteed. Thus, this assumption limits the effective application of these algorithms in practice. This work relaxes this restrictive assumption while still obtaining similar results under weaker assumptions, thereby guaranteeing effectiveness of these algorithms. For the setting under consideration, it is shown for vector fields on compact Riemannian manifolds that the restrictive assumption follows immediately and does not need to be independently verified. A motivating example shows that this is not the case for vector fields on Euclidean space, but in this setting it is shown that the restrictive assumption can still be relaxed provided there exist a neighborhood of infinity with suitable properties and some additional generic assumptions. These results are then used to provide theoretical guarantees for the numerical algorithms discussed above under far weaker assumptions.
{"title":"Stability of the Nonwandering Set in the Region of Attraction Boundary under Perturbations with Application to Vulnerability Assessment","authors":"Michael W. Fisher, Ian A. Hiskens","doi":"10.1137/23m155582x","DOIUrl":"https://doi.org/10.1137/23m155582x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3390-3430, December 2023. <br/> Abstract. For many engineered systems it is important to assess vulnerability to potential disturbances in order to ensure reliable operation. Whether the system will recover from a particular finite-time disturbance to a desired stable equilibrium point depends on uncertain and time-varying system parameter values. Therefore, it is valuable to determine, for specific fixed disturbances, the margins for safe operation: the smallest change in parameter values that would cause the system to become vulnerable to the disturbance. The natural setting for this problem is a parameter-dependent vector field with a family of stable equilibria and a parameter-dependent initial condition representing the disturbance. The system recovers for a particular parameter value if its initial condition lies within the region of attraction of the desired stable equilibrium point. Prior work has developed algorithms for numerically computing the margins for safe operation. However, the theoretical guarantees provided for these methods require a very restrictive assumption: that the nonwandering set in the region of attraction boundary is stable under perturbations to the vector field. This assumption is generally intractable to verify, so feasibility of the above algorithms cannot be determined in advance, and even when these algorithms do converge their convergence to the correct values cannot be guaranteed. Thus, this assumption limits the effective application of these algorithms in practice. This work relaxes this restrictive assumption while still obtaining similar results under weaker assumptions, thereby guaranteeing effectiveness of these algorithms. For the setting under consideration, it is shown for vector fields on compact Riemannian manifolds that the restrictive assumption follows immediately and does not need to be independently verified. A motivating example shows that this is not the case for vector fields on Euclidean space, but in this setting it is shown that the restrictive assumption can still be relaxed provided there exist a neighborhood of infinity with suitable properties and some additional generic assumptions. These results are then used to provide theoretical guarantees for the numerical algorithms discussed above under far weaker assumptions.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138552725","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 22, Issue 4, Page 3320-3357, December 2023. Abstract. A semianalytical method is derived for finding the existence and stability of single-impact periodic orbits born in a boundary equilibrium bifurcation in a general [math]-dimensional impacting hybrid system. Known results are reproduced for planar systems and general formulae derived for three-dimensional (3D) systems. A numerical implementation of the method is illustrated for several 3D examples and for an 8D wing-flap model that shows coexistence of attractors. It is shown how the method can easily be embedded within numerical continuation, and some remarks are made about necessary and sufficient conditions in arbitrary dimensional systems.
{"title":"Bifurcation of Limit Cycles from Boundary Equilibria in Impacting Hybrid Systems","authors":"Hong Tang, Alan Champneys","doi":"10.1137/23m1552292","DOIUrl":"https://doi.org/10.1137/23m1552292","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3320-3357, December 2023. <br/> Abstract. A semianalytical method is derived for finding the existence and stability of single-impact periodic orbits born in a boundary equilibrium bifurcation in a general [math]-dimensional impacting hybrid system. Known results are reproduced for planar systems and general formulae derived for three-dimensional (3D) systems. A numerical implementation of the method is illustrated for several 3D examples and for an 8D wing-flap model that shows coexistence of attractors. It is shown how the method can easily be embedded within numerical continuation, and some remarks are made about necessary and sufficient conditions in arbitrary dimensional systems.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"35 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138545624","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 22, Issue 4, Page 3358-3389, December 2023. Abstract. The rocking can problem [M. Srinivasan and A. Ruina, Phys. Rev. E, 78 (2008), 066609] consists of an empty drinks can standing upright on a horizontal plane which, when tipped back to a single contact point and released, rocks down towards the flat and level state. At the bottom of the motion, the contact point moves quickly around the rim of the can. The can then rises up again, having rotated through some finite angle of turn [math]. We recast the problem as a second order ODE and find a Frobenius solution. We then use this Frobenius solution to derive a reduced equation of motion. The rocking can exhibits two distinct phenomena: behavior very similar to an inverted pendulum, and dynamics with the angle of turn. This distinction allows us to use matched asymptotic expansions to derive a uniformly valid solution that is in excellent agreement with numerical calculations of the reduced equation of motion. The solution of the inner problem was used to investigate of the angle of turn phenomenon. We also examine the motion of the contact locus [math] and see a range of different trajectories, from circular to petaloid motion and even cusp-like behavior. Finally, we obtain an approximate lower bound for the required coefficient of friction to avoid slip.
SIAM 应用动力系统期刊》，第 22 卷第 4 期，第 3358-3389 页，2023 年 12 月。 摘要。摇罐问题 [M. Srinivasan and A. Ruina, Phys. Rev. E, 78 (2008), 066609] 包含一个直立在水平面上的空饮料罐。在运动的底部，接触点围绕罐子边缘快速移动。然后，罐子再次上升，旋转了某个有限的角度[数学]。我们将问题重塑为一个二阶 ODE，并找到一个 Frobenius 解。然后，我们利用这个弗罗贝尼斯解推导出一个简化的运动方程。摇摆可以表现出两种截然不同的现象：与倒立摆非常相似的行为，以及随转角变化的动力学。这种区别使我们能够使用匹配的渐近展开求出统一有效的解，该解与简化运动方程的数值计算结果非常吻合。内部问题的解被用来研究转角现象。我们还研究了接触点的运动[math]，看到了一系列不同的轨迹，从圆周运动到花瓣运动，甚至是类似尖顶的行为。最后，我们得到了避免滑移所需的摩擦系数的近似下限。
{"title":"The Rocking Can: A Reduced Equation of Motion and a Matched Asymptotic Solution","authors":"B. W. Collins, C. L. Hall, S. J. Hogan","doi":"10.1137/23m1551031","DOIUrl":"https://doi.org/10.1137/23m1551031","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3358-3389, December 2023. <br/> Abstract. The rocking can problem [M. Srinivasan and A. Ruina, Phys. Rev. E, 78 (2008), 066609] consists of an empty drinks can standing upright on a horizontal plane which, when tipped back to a single contact point and released, rocks down towards the flat and level state. At the bottom of the motion, the contact point moves quickly around the rim of the can. The can then rises up again, having rotated through some finite angle of turn [math]. We recast the problem as a second order ODE and find a Frobenius solution. We then use this Frobenius solution to derive a reduced equation of motion. The rocking can exhibits two distinct phenomena: behavior very similar to an inverted pendulum, and dynamics with the angle of turn. This distinction allows us to use matched asymptotic expansions to derive a uniformly valid solution that is in excellent agreement with numerical calculations of the reduced equation of motion. The solution of the inner problem was used to investigate of the angle of turn phenomenon. We also examine the motion of the contact locus [math] and see a range of different trajectories, from circular to petaloid motion and even cusp-like behavior. Finally, we obtain an approximate lower bound for the required coefficient of friction to avoid slip.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"133 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138545842","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 22, Issue 4, Page 3284-3319, December 2023. Abstract. The intention of this article is to illustrate the use of methods from symplectic geometry for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g., the three-body problem). The main directions pursued in this article are as follows: (1) given two periodic orbits, decide when they can be connected by a regular family of periodic orbits; (2) use numerical invariants from Floer theory which help predict the existence of orbits in the presence of a bifurcation; (3) attach a sign [math] to each elliptic or hyperbolic Floquet multiplier of a closed symmetric orbit, which generalizes the classical Krein–Moser sign to also include the hyperbolic case; and (4) do all of the above in a visual, easily implementable, and resource-efficient way. The mathematical framework is provided by the first and third authors in [U. Frauenfelder and A. Moreno, J. Symplectic Geom., to appear], where, as it turns out, the “Broucke stability diagram” [R. Broucke, AIAA J., 7 (1969), pp. 1003–1009] was rediscovered, but further refined with the above signs and algebraically reformulated in terms of quotients of the symplectic group. The advantage of the framework is that it applies to the study of closed orbits of an arbitrary Hamiltonian system. We will carry out numerical work based on the cell-mapping method as described in [D. Koh, R. L. Anderson, and I. Bermejo-Moreno, J. Astronautical Sci., 68 (2021), pp. 172–196] for the Jupiter-Europa and Saturn-Enceladus systems. These are currently systems of interest, falling in the agenda of space agencies like NASA, as these icy moons are considered candidates for harboring conditions suitable for extraterrestrial life.
应用动力系统学报，第22卷，第4期，3284-3319页，2023年12月。摘要。本文的目的是说明辛几何方法的实际用途。我们的目标读者是对哈密顿系统的轨道(例如，三体问题)感兴趣的科学家。本文的主要研究方向如下:(1)给定两个周期轨道，确定它们何时可以由一个规则的周期轨道族连接起来;(2)利用Floer理论中的数值不变量来预测存在分岔时轨道的存在性;(3)在闭对称轨道的每个椭圆或双曲Floquet乘法器上附加一个符号[math]，将经典的klein - moser符号推广到双曲情况;(4)以可视化、易于实现和资源高效的方式完成上述所有工作。数学框架是由[美国]的第一和第三作者提供的。莫雷诺，李建平。，在那里，事实证明，“布鲁克稳定性图”[R。Broucke, AIAA J.， 7 (1969)， pp. 1003-1009]被重新发现，但进一步改进了上述符号，并用辛群的商进行了代数上的重新表述。该框架的优点是它适用于研究任意哈密顿系统的封闭轨道。我们将根据[D]中描述的细胞映射方法开展数值工作。Koh, R. L. Anderson, I. Bermejo-Moreno, J.宇航科学。木星-木卫二和土星-土卫二系统，68(2021)，第172-196页。这些都是目前人们感兴趣的系统，被列入了美国宇航局等太空机构的议程，因为这些冰冷的卫星被认为是适合外星生命生存的候选者。
{"title":"Symplectic Methods in the Numerical Search of Orbits in Real-Life Planetary Systems","authors":"Urs Frauenfelder, Dayung Koh, Agustin Moreno","doi":"10.1137/22m1500459","DOIUrl":"https://doi.org/10.1137/22m1500459","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3284-3319, December 2023. <br/> Abstract. The intention of this article is to illustrate the use of methods from symplectic geometry for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g., the three-body problem). The main directions pursued in this article are as follows: (1) given two periodic orbits, decide when they can be connected by a regular family of periodic orbits; (2) use numerical invariants from Floer theory which help predict the existence of orbits in the presence of a bifurcation; (3) attach a sign [math] to each elliptic or hyperbolic Floquet multiplier of a closed symmetric orbit, which generalizes the classical Krein–Moser sign to also include the hyperbolic case; and (4) do all of the above in a visual, easily implementable, and resource-efficient way. The mathematical framework is provided by the first and third authors in [U. Frauenfelder and A. Moreno, J. Symplectic Geom., to appear], where, as it turns out, the “Broucke stability diagram” [R. Broucke, AIAA J., 7 (1969), pp. 1003–1009] was rediscovered, but further refined with the above signs and algebraically reformulated in terms of quotients of the symplectic group. The advantage of the framework is that it applies to the study of closed orbits of an arbitrary Hamiltonian system. We will carry out numerical work based on the cell-mapping method as described in [D. Koh, R. L. Anderson, and I. Bermejo-Moreno, J. Astronautical Sci., 68 (2021), pp. 172–196] for the Jupiter-Europa and Saturn-Enceladus systems. These are currently systems of interest, falling in the agenda of space agencies like NASA, as these icy moons are considered candidates for harboring conditions suitable for extraterrestrial life.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"46 ","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505527","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 22, Issue 4, Page 3267-3283, December 2023. Abstract. We prove that the Kuramoto model on a graph can contain infinitely many nonequivalent stable equilibria. More precisely, we prove that for every [math] there is a connected graph such that the set of stable equilibria contains a manifold of dimension [math]. In particular, we solve a conjecture of Delabays, Coletta, and Jacquod about the number of equilibria on planar graphs. Our results are based on the analysis of balanced configurations, which correspond to equilateral polygon linkages in topology. In order to analyze the stability of manifolds of equilibria we apply topological bifurcation theory.
应用动力系统学报，vol . 22, Issue 4, Page 3267-3283, December 2023。摘要。证明了图上的Kuramoto模型可以包含无穷多个非等价稳定平衡点。更准确地说，我们证明了对于每一个[数学]存在一个连通图，使得稳定均衡集包含一个维数[数学]的流形。特别地，我们解决了Delabays, Coletta和Jacquod关于平面图上平衡点数目的猜想。我们的结果是基于平衡构型的分析，它对应于拓扑中的等边多边形连杆。为了分析平衡流形的稳定性，我们应用了拓扑分岔理论。
{"title":"Kuramoto Networks with Infinitely Many Stable Equilibria","authors":"Davide Sclosa","doi":"10.1137/23m155400x","DOIUrl":"https://doi.org/10.1137/23m155400x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3267-3283, December 2023. <br/> Abstract. We prove that the Kuramoto model on a graph can contain infinitely many nonequivalent stable equilibria. More precisely, we prove that for every [math] there is a connected graph such that the set of stable equilibria contains a manifold of dimension [math]. In particular, we solve a conjecture of Delabays, Coletta, and Jacquod about the number of equilibria on planar graphs. Our results are based on the analysis of balanced configurations, which correspond to equilateral polygon linkages in topology. In order to analyze the stability of manifolds of equilibria we apply topological bifurcation theory.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"44 11","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505513","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}
Electa Cleveland, Angela Zhu, Björn Sandstede, Alexandria Volkening
SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3233-3266, December 2023. Abstract. Mathematical models come in many forms across biological applications. In the case of complex, spatial dynamics and pattern formation, stochastic models also face two main challenges: pattern data are largely qualitative, and model realizations may vary significantly. Together these issues make it difficult to relate models and empirical data—or even models and models—limiting how different approaches can be combined to offer new insights into biology. These challenges also raise mathematical questions about how models are related, since alternative approaches to the same problem—e.g., Cellular Potts models; off-lattice, agent-based models; on-lattice, cellular automaton models; and continuum approaches—treat uncertainty and implement cell behavior in different ways. To help open the door to future work on questions like these, here we adapt methods from topological data analysis and computational geometry to quantitatively relate two different models of the same biological process in a fair, comparable way. To center our work and illustrate concrete challenges, we focus on the example of zebrafish-skin pattern formation, and we relate patterns that arise from agent-based and cellular automaton models.
{"title":"Quantifying Different Modeling Frameworks Using Topological Data Analysis: A Case Study with Zebrafish Patterns","authors":"Electa Cleveland, Angela Zhu, Björn Sandstede, Alexandria Volkening","doi":"10.1137/22m1543082","DOIUrl":"https://doi.org/10.1137/22m1543082","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3233-3266, December 2023. <br/> Abstract. Mathematical models come in many forms across biological applications. In the case of complex, spatial dynamics and pattern formation, stochastic models also face two main challenges: pattern data are largely qualitative, and model realizations may vary significantly. Together these issues make it difficult to relate models and empirical data—or even models and models—limiting how different approaches can be combined to offer new insights into biology. These challenges also raise mathematical questions about how models are related, since alternative approaches to the same problem—e.g., Cellular Potts models; off-lattice, agent-based models; on-lattice, cellular automaton models; and continuum approaches—treat uncertainty and implement cell behavior in different ways. To help open the door to future work on questions like these, here we adapt methods from topological data analysis and computational geometry to quantitatively relate two different models of the same biological process in a fair, comparable way. To center our work and illustrate concrete challenges, we focus on the example of zebrafish-skin pattern formation, and we relate patterns that arise from agent-based and cellular automaton models.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"47 ","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505526","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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