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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3208-3232, December 2023. Abstract.The deep linear network (DLN) is a model for implicit regularization in gradient based optimization of overparametrized learning architectures. Training the DLN corresponds to a Riemannian gradient flow, where the Riemannian metric is defined by the architecture of the network and the loss function is defined by the learning task. We extend this geometric framework, obtaining explicit expressions for the volume form, including the case when the network has infinite depth. We investigate the link between the Riemannian geometry and the training asymptotics for matrix completion with rigorous analysis and numerics. We propose that under small initialization, implicit regularization is a result of bias towards high state space volume.
应用动力系统学报，第22卷，第4期，第3208-3232页，2023年12月。摘要。深度线性网络(deep linear network, DLN)是一种基于梯度优化的隐式正则化模型。训练DLN对应于黎曼梯度流，其中黎曼度量由网络的体系结构定义，损失函数由学习任务定义。我们扩展了这个几何框架，得到了体积形式的显式表达式，包括网络具有无限深度的情况。我们用严格的分析和数值研究了黎曼几何和矩阵补全的训练渐近之间的联系。我们提出在小初始化下，隐式正则化是偏向于高状态空间体积的结果。
{"title":"Deep Linear Networks for Matrix Completion—an Infinite Depth Limit","authors":"Nadav Cohen, Govind Menon, Zsolt Veraszto","doi":"10.1137/22m1530653","DOIUrl":"https://doi.org/10.1137/22m1530653","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3208-3232, December 2023. <br/> Abstract.The deep linear network (DLN) is a model for implicit regularization in gradient based optimization of overparametrized learning architectures. Training the DLN corresponds to a Riemannian gradient flow, where the Riemannian metric is defined by the architecture of the network and the loss function is defined by the learning task. We extend this geometric framework, obtaining explicit expressions for the volume form, including the case when the network has infinite depth. We investigate the link between the Riemannian geometry and the training asymptotics for matrix completion with rigorous analysis and numerics. We propose that under small initialization, implicit regularization is a result of bias towards high state space volume.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505528","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 3165-3207, December 2023. Abstract. In an epidemic network, lags due to travel time between populations, latent period, and recovery period can significantly change the epidemic behavior and result in successive echoing waves of the spread between various population clusters. Moreover, external shocks to a given population can propagate to other populations within the network, potentially snowballing into waves of resurgent epidemics. The main objective of this study is to investigate the effect of time delay and small shocks/uncertainties on the linear susceptible-infectious-susceptible (SIS) dynamics of epidemic networks. In this regard, the asymptotic stability of this class of networks is first studied, and then its performance loss due to small shocks/uncertainties is evaluated based on the notion of the [math] norm. It is shown that network performance loss is correlated with the structure of the underlying graph, intrinsic time delays, epidemic characteristics, and external shocks. This performance measure is then used to develop an optimal traffic restriction algorithm for network performance enhancement, resulting in reduced infection in the metapopulation. A novel epidemic-based centrality index is also defined to evaluate the impact of every subpopulation on network performance, and its asymptotic behavior is investigated. It is shown that for specific choices of parameters, the output of the epidemic-based centrality index converges to the results obtained by local or eigenvector centralities. Moreover, given that epidemic-based centrality depends on the epidemic properties of the disease, it may yield distinct node rankings as the disease characteristics slowly change over time or as different types of infections spread. This node interlacing phenomenon is not observed in other centralities that rely solely on network structure. This unique characteristic of epidemic-based centrality enables it to adjust to various epidemic features. The derived centrality index is then adopted to improve the network robustness against external shocks on the epidemic network. The numerical results, along with the theoretical expectations, highlight the role of time delay as well as small shocks in investigating the most effective methods of epidemic containment.
应用动力系统学报，vol . 22, Issue 4, Page 3165-3207, December 2023。摘要。在疫情网络中，由于种群间的传播时间、潜伏期和恢复期的滞后会显著改变疫情行为，导致不同种群间传播的连续回声波。此外，对某一特定人群的外部冲击可能会传播到网络内的其他人群，从而可能滚雪球般地形成一波又一波的流行病。本研究的主要目的是研究时间延迟和小冲击/不确定性对流行病网络线性易感-感染-易感(SIS)动力学的影响。在这方面，首先研究了这类网络的渐近稳定性，然后基于[数学]范数的概念评估了其由于小冲击/不确定性而造成的性能损失。结果表明，网络性能损失与底层图的结构、固有时滞、流行特征和外部冲击有关。然后使用该性能度量来开发网络性能增强的最优流量限制算法，从而减少元种群中的感染。定义了一种新的基于流行度的中心性指数来评价每个子种群对网络性能的影响，并研究了其渐近行为。结果表明，对于特定参数的选择，基于流行病的中心性指数的输出收敛于局部中心性或特征向量中心性得到的结果。此外，鉴于基于流行病的中心性取决于疾病的流行病特性，随着疾病特征随时间缓慢变化或不同类型的感染传播，它可能产生不同的节点排名。这种节点交错现象在其他仅依赖于网络结构的中心性中没有观察到。这种基于流行病的中心性的独特特征使其能够适应各种流行病特征。然后采用导出的中心性指数来提高网络对疫情网络外部冲击的鲁棒性。数值结果以及理论期望突出了时间延迟和小冲击在研究最有效的流行病控制方法中的作用。
{"title":"Centrality-Based Traffic Restriction in Delayed Epidemic Networks","authors":"Atefe Darabi, Milad Siami","doi":"10.1137/22m1507760","DOIUrl":"https://doi.org/10.1137/22m1507760","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3165-3207, December 2023. <br/> Abstract. In an epidemic network, lags due to travel time between populations, latent period, and recovery period can significantly change the epidemic behavior and result in successive echoing waves of the spread between various population clusters. Moreover, external shocks to a given population can propagate to other populations within the network, potentially snowballing into waves of resurgent epidemics. The main objective of this study is to investigate the effect of time delay and small shocks/uncertainties on the linear susceptible-infectious-susceptible (SIS) dynamics of epidemic networks. In this regard, the asymptotic stability of this class of networks is first studied, and then its performance loss due to small shocks/uncertainties is evaluated based on the notion of the [math] norm. It is shown that network performance loss is correlated with the structure of the underlying graph, intrinsic time delays, epidemic characteristics, and external shocks. This performance measure is then used to develop an optimal traffic restriction algorithm for network performance enhancement, resulting in reduced infection in the metapopulation. A novel epidemic-based centrality index is also defined to evaluate the impact of every subpopulation on network performance, and its asymptotic behavior is investigated. It is shown that for specific choices of parameters, the output of the epidemic-based centrality index converges to the results obtained by local or eigenvector centralities. Moreover, given that epidemic-based centrality depends on the epidemic properties of the disease, it may yield distinct node rankings as the disease characteristics slowly change over time or as different types of infections spread. This node interlacing phenomenon is not observed in other centralities that rely solely on network structure. This unique characteristic of epidemic-based centrality enables it to adjust to various epidemic features. The derived centrality index is then adopted to improve the network robustness against external shocks on the epidemic network. The numerical results, along with the theoretical expectations, highlight the role of time delay as well as small shocks in investigating the most effective methods of epidemic containment.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505525","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}
Tea Vojkovic, David Quero, Charles Poussot-Vassal, Pierre Vuillemin
SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3130-3164, December 2023. Abstract.In this work, we present a novel data-driven method for identifying parametric MIMO generalized state-space or descriptor systems of low order that accurately capture the frequency and time domain behavior of large-scale linear dynamical systems. The low-order parametric descriptor systems are identified from transfer matrix samples by means of two-variable Lagrange rational matrix interpolation. This is done within the Loewner framework by deploying the new matrix-valued barycentric formula given in both right and left polynomial matrix fraction forms, which enables the construction of minimal parametric descriptor systems with rectangular transfer matrices. The developed method allows the reduction of order and parameter dependence complexity of the constructed system. Stability of the system is preserved by the postprocessing technique based on flipping signs of unstable poles. The developed methodology is illustrated with a few academic examples and applied to low-order parametric state-space identification of an aerodynamic system.
{"title":"Low-Order Parametric State-Space Modeling of MIMO Systems in the Loewner Framework","authors":"Tea Vojkovic, David Quero, Charles Poussot-Vassal, Pierre Vuillemin","doi":"10.1137/22m1509898","DOIUrl":"https://doi.org/10.1137/22m1509898","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3130-3164, December 2023. <br/> Abstract.In this work, we present a novel data-driven method for identifying parametric MIMO generalized state-space or descriptor systems of low order that accurately capture the frequency and time domain behavior of large-scale linear dynamical systems. The low-order parametric descriptor systems are identified from transfer matrix samples by means of two-variable Lagrange rational matrix interpolation. This is done within the Loewner framework by deploying the new matrix-valued barycentric formula given in both right and left polynomial matrix fraction forms, which enables the construction of minimal parametric descriptor systems with rectangular transfer matrices. The developed method allows the reduction of order and parameter dependence complexity of the constructed system. Stability of the system is preserved by the postprocessing technique based on flipping signs of unstable poles. The developed methodology is illustrated with a few academic examples and applied to low-order parametric state-space identification of an aerodynamic system.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505529","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}
Olivier Hénot, Jean-Philippe Lessard, Jason D. Mireles James
SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3093-3129, December 2023. Abstract. We present a computational method for studying transverse homoclinic orbits for periodic solutions of delay differential equations, a phenomenon that we refer to as the Poincaré scenario. The strategy is geometric in nature and consists of viewing the connection as the zero of a nonlinear map, such that the invertibility of its Fréchet derivative implies the transversality of the intersection. The map is defined by a projected boundary value problem (BVP), with boundary conditions in the (finite dimensional) unstable and (infinite dimensional) stable manifolds of the periodic orbit. The parameterization method is used to compute the unstable manifold, and the BVP is solved using a discrete time dynamical system approach (defined via the method of steps) and Chebyshev series expansions. We illustrate this technique by computing transverse homoclinic orbits in the cubic Ikeda and Mackey–Glass systems.
{"title":"Numerical Computation of Transverse Homoclinic Orbits for Periodic Solutions of Delay Differential Equations","authors":"Olivier Hénot, Jean-Philippe Lessard, Jason D. Mireles James","doi":"10.1137/23m1562858","DOIUrl":"https://doi.org/10.1137/23m1562858","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3093-3129, December 2023. <br/> Abstract. We present a computational method for studying transverse homoclinic orbits for periodic solutions of delay differential equations, a phenomenon that we refer to as the Poincaré scenario. The strategy is geometric in nature and consists of viewing the connection as the zero of a nonlinear map, such that the invertibility of its Fréchet derivative implies the transversality of the intersection. The map is defined by a projected boundary value problem (BVP), with boundary conditions in the (finite dimensional) unstable and (infinite dimensional) stable manifolds of the periodic orbit. The parameterization method is used to compute the unstable manifold, and the BVP is solved using a discrete time dynamical system approach (defined via the method of steps) and Chebyshev series expansions. We illustrate this technique by computing transverse homoclinic orbits in the cubic Ikeda and Mackey–Glass systems.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505522","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}
The spectrum of the Koopman operator has been shown to encode many important statistical and geometric properties of a dynamical system. In this work, we consider induced linear operators acting on the space of sections of the state space’s tangent, cotangent, and tensor bundles. We first demonstrate how these operators are natural generalizations of Koopman operators acting on functions. We then draw connections between the various operators’ spectra and characterize the algebraic and differential topological properties of their spectra. We describe the discrete spectrum of these operators for linear dynamical systems and derive spectral expansions for linear vector fields. We define the notion of an “eigendistribution,” provide conditions for an eigendistribution to be integrable, and demonstrate how to recover the foliations arising from their integral manifolds. Last, we demonstrate that the characteristic Lyapunov exponents of a uniformly hyperbolic dynamical system are in the spectrum of the induced operators on sections of the tangent or cotangent bundle. We conclude with an application to differential geometry where the well-known fact that the flows of commuting vector fields commute is generalized, and we recover the original statement as a particular case of our result. We also apply our results to recover the Lyapunov exponents and the stable/unstable foliations of Arnold’s cat map via the spectrum of the induced operator on sections of the tangent bundle.
{"title":"Spectral Properties of Pullback Operators on Vector Bundles of a Dynamical System","authors":"Allan M. Avila, Igor Mezić","doi":"10.1137/22m1492064","DOIUrl":"https://doi.org/10.1137/22m1492064","url":null,"abstract":"The spectrum of the Koopman operator has been shown to encode many important statistical and geometric properties of a dynamical system. In this work, we consider induced linear operators acting on the space of sections of the state space’s tangent, cotangent, and tensor bundles. We first demonstrate how these operators are natural generalizations of Koopman operators acting on functions. We then draw connections between the various operators’ spectra and characterize the algebraic and differential topological properties of their spectra. We describe the discrete spectrum of these operators for linear dynamical systems and derive spectral expansions for linear vector fields. We define the notion of an “eigendistribution,” provide conditions for an eigendistribution to be integrable, and demonstrate how to recover the foliations arising from their integral manifolds. Last, we demonstrate that the characteristic Lyapunov exponents of a uniformly hyperbolic dynamical system are in the spectrum of the induced operators on sections of the tangent or cotangent bundle. We conclude with an application to differential geometry where the well-known fact that the flows of commuting vector fields commute is generalized, and we recover the original statement as a particular case of our result. We also apply our results to recover the Lyapunov exponents and the stable/unstable foliations of Arnold’s cat map via the spectrum of the induced operator on sections of the tangent bundle.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135286406","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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