Brendan Harding, Yvonne M. Stokes, Rahil N. Valani
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1805-1835, September 2024. Abstract.Inertial focusing in curved microfluidic ducts exploits the interaction of the drag force from the Dean flow with the inertial lift force to separate particles or cells laterally across the cross-section width according to their size. Experimental work has identified that using a trapezoidal cross section, as opposed to a rectangular one, can enhance the sized based separation of particles/cells over a wide range of flow rates. Using our model, derived by carefully examining the way the Dean drag and inertial lift forces interact at low flow rates, we calculate the leading order approximation of these forces for a range of trapezoidal ducts, both vertically symmetric and nonsymmetric, with an increasing amount of skew towards the outside wall. We then conduct a systematic study to examine the bifurcations in the particle equilbira that occur with respect to a shape parameter characterizing the trapezoidal cross section. We reveal how the dynamics associated with particle migration are modified by the degree of skew in the cross-section shape, and show the existence of cusp bifurcations (with the bend radius as a second parameter). Additionally, our investigation suggests an optimal amount of skew for the trapezoidal cross section for the purposes of maximizing particle separation over a wide range of bend radii.
{"title":"Inertial Focusing Dynamics of Spherical Particles in Curved Microfluidic Ducts with a Trapezoidal Cross Section","authors":"Brendan Harding, Yvonne M. Stokes, Rahil N. Valani","doi":"10.1137/23m1613220","DOIUrl":"https://doi.org/10.1137/23m1613220","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1805-1835, September 2024. <br/> Abstract.Inertial focusing in curved microfluidic ducts exploits the interaction of the drag force from the Dean flow with the inertial lift force to separate particles or cells laterally across the cross-section width according to their size. Experimental work has identified that using a trapezoidal cross section, as opposed to a rectangular one, can enhance the sized based separation of particles/cells over a wide range of flow rates. Using our model, derived by carefully examining the way the Dean drag and inertial lift forces interact at low flow rates, we calculate the leading order approximation of these forces for a range of trapezoidal ducts, both vertically symmetric and nonsymmetric, with an increasing amount of skew towards the outside wall. We then conduct a systematic study to examine the bifurcations in the particle equilbira that occur with respect to a shape parameter characterizing the trapezoidal cross section. We reveal how the dynamics associated with particle migration are modified by the degree of skew in the cross-section shape, and show the existence of cusp bifurcations (with the bend radius as a second parameter). Additionally, our investigation suggests an optimal amount of skew for the trapezoidal cross section for the purposes of maximizing particle separation over a wide range of bend radii.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609967","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 3, Page 1766-1804, September 2024. Abstract. This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles (and systems of such circles) in area preserving maps. The recipe requires only a finite data set sampled from the quasiperiodic circle. Our approach, being based on the parameterization method of [A. Haro and R. de la Llave, SIAM J. Appl. Dyn. Syst., 6 (2007), pp. 142–207; A. Haro and R. de la Llave, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261–1300; A. Haro and R. de la Llave, J. Differential Equations, 228 (2006), pp. 530–579], uses a Newton scheme to iteratively solve a conjugacy equation describing the invariant circle (or systems of circles). A critical step in properly formulating the conjugacy equation is to determine the rotation number of the quasiperiodic subsystem. For this we exploit the weighted Birkhoff averaging method of [S. Das et al., Nonlinearity, 30 (2017), pp. 4111–4140; S. Das et al., The Foundations of Chaos Revisited, Springer, Cham, 2016, pp. 103–118; S. Das and J. A. Yorke, Nonlinearity, 31 (2018), pp. 491–501]. This approach facilities accurate computation of the rotation number given nothing but the already mentioned orbit data. The weighted Birkhoff averages also facilitate the computation of other integral observables like Fourier coefficients of the parameterization of the invariant circle. Since, the parameterization method is based on a Newton scheme, we only need to approximate a small number of Fourier coefficients with low accuracy (say, a few correct digits) to find a good enough initial approximation so that Newton converges. Moreover, the Fourier coefficients may be computed independently, so we can sample the higher modes to guess the decay rate of the Fourier coefficients. This allows us to choose, a priori, an appropriate number of modes in the truncation. We illustrate the utility of the approach for explicit example systems including the area preserving Hénon map and the standard map (polynomial and trigonometric nonlinearity respectively). We present example computations for invariant circles and for systems of invariant circles with as many as 120 components. We also employ a numerical continuation scheme (where the rotation number is the continuation parameter) to compute large numbers of quasiperiodic circles in these systems. During the continuation we monitor the Sobolev norm of the parameterization, as explained in [R. Calleja and R. de la Llave, Nonlinearity, 23 (2010), pp. 2029–2058], to automatically detect the breakdown of the family.
SIAM 应用动力系统期刊》，第 23 卷第 3 期，第 1766-1804 页，2024 年 9 月。 摘要。本研究提供了一种系统的方法，用于计算面积保留映射中准周期不变圆（以及此类圆的系统）的精确高阶傅里叶展开。该方法只需要从准周期圆中采样的有限数据集。我们的方法基于[A. Haro and R. de la LL]的参数化方法。Haro and R. de la Llave, SIAM J. Appl.Syst., 6 (2007), pp.Dyn.Syst.B, 6 (2006), pp. 1261-1300; A. Haro and R. de la Llave, J. Differential Equations, 228 (2006), pp.正确表述共轭方程的关键步骤是确定准周期子系统的旋转数。为此，我们利用了加权伯克霍夫平均法 [S. Das et al.S. Das 等人，《非线性》，30 (2017)，第 4111-4140 页；S. Das 等人，《混沌基础重温》，Springer, Cham, 2016, 第 103-118 页；S. Das 和 J. A. Yorke，《非线性》，31 (2018)，第 491-501 页]。除了上述轨道数据，这种方法还能准确计算旋转数。加权伯克霍夫平均数还有助于计算其他积分观测值，如不变圆参数化的傅里叶系数。由于参数化方法以牛顿方案为基础，我们只需对少量傅里叶系数进行低精度近似（例如几位正确数字），就能找到足够好的初始近似值，从而使牛顿收敛。此外，傅立叶系数可以独立计算，因此我们可以对高阶模式进行采样，从而猜测傅立叶系数的衰减率。这样，我们就可以先验地选择适当数量的截断模式。我们将举例说明这种方法对包括面积保持的赫农图谱和标准图谱（分别为多项式和三角非线性）在内的显式系统的实用性。我们举例说明了不变圆和多达 120 个分量的不变圆系统的计算。我们还采用数值延续方案（其中旋转数是延续参数）来计算这些系统中的大量准周期圆。在延续过程中，我们按照 [R. Calleja and R. de la Llave, Nonlinearity, 23 (2010), pp.
{"title":"Weighted Birkhoff Averages and the Parameterization Method","authors":"David Blessing, J. D. Mireles James","doi":"10.1137/23m1579546","DOIUrl":"https://doi.org/10.1137/23m1579546","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1766-1804, September 2024. <br/> Abstract. This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles (and systems of such circles) in area preserving maps. The recipe requires only a finite data set sampled from the quasiperiodic circle. Our approach, being based on the parameterization method of [A. Haro and R. de la Llave, SIAM J. Appl. Dyn. Syst., 6 (2007), pp. 142–207; A. Haro and R. de la Llave, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261–1300; A. Haro and R. de la Llave, J. Differential Equations, 228 (2006), pp. 530–579], uses a Newton scheme to iteratively solve a conjugacy equation describing the invariant circle (or systems of circles). A critical step in properly formulating the conjugacy equation is to determine the rotation number of the quasiperiodic subsystem. For this we exploit the weighted Birkhoff averaging method of [S. Das et al., Nonlinearity, 30 (2017), pp. 4111–4140; S. Das et al., The Foundations of Chaos Revisited, Springer, Cham, 2016, pp. 103–118; S. Das and J. A. Yorke, Nonlinearity, 31 (2018), pp. 491–501]. This approach facilities accurate computation of the rotation number given nothing but the already mentioned orbit data. The weighted Birkhoff averages also facilitate the computation of other integral observables like Fourier coefficients of the parameterization of the invariant circle. Since, the parameterization method is based on a Newton scheme, we only need to approximate a small number of Fourier coefficients with low accuracy (say, a few correct digits) to find a good enough initial approximation so that Newton converges. Moreover, the Fourier coefficients may be computed independently, so we can sample the higher modes to guess the decay rate of the Fourier coefficients. This allows us to choose, a priori, an appropriate number of modes in the truncation. We illustrate the utility of the approach for explicit example systems including the area preserving Hénon map and the standard map (polynomial and trigonometric nonlinearity respectively). We present example computations for invariant circles and for systems of invariant circles with as many as 120 components. We also employ a numerical continuation scheme (where the rotation number is the continuation parameter) to compute large numbers of quasiperiodic circles in these systems. During the continuation we monitor the Sobolev norm of the parameterization, as explained in [R. Calleja and R. de la Llave, Nonlinearity, 23 (2010), pp. 2029–2058], to automatically detect the breakdown of the family.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585295","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 3, Page 1720-1765, September 2024. Abstract.This paper is to investigate synchronization theories of a two-group Kuramoto model and a three-group Kuramoto model. In the settings of these models, every oscillator directly interacts with each other in the same group. In each group, only one oscillator directly interacts with one oscillator in another group. We prove that if the coupling strength is large and the initial configuration of each group is confined to a sector with the arc length less than [math], then all oscillators achieve a complete frequency synchronization asymptotically. We emphasize that there is no need to impose any initial condition on the connection between different groups. If, in addition, the natural frequencies in one group are the same, then partial phase synchronization occurs. Moreover, if all natural frequencies are identical, we prove that all oscillators either achieve a complete phase synchronization asymptotically or tend to a bipolar phase-locking state. We also provide several numerical simulations to support the main results.
{"title":"Complete and Partial Synchronization of Two-Group and Three-Group Kuramoto Oscillators","authors":"Shih-Hsin Chen, Chun-Hsiung Hsia, Ting-Yang Hsiao","doi":"10.1137/23m1586227","DOIUrl":"https://doi.org/10.1137/23m1586227","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1720-1765, September 2024. <br/> Abstract.This paper is to investigate synchronization theories of a two-group Kuramoto model and a three-group Kuramoto model. In the settings of these models, every oscillator directly interacts with each other in the same group. In each group, only one oscillator directly interacts with one oscillator in another group. We prove that if the coupling strength is large and the initial configuration of each group is confined to a sector with the arc length less than [math], then all oscillators achieve a complete frequency synchronization asymptotically. We emphasize that there is no need to impose any initial condition on the connection between different groups. If, in addition, the natural frequencies in one group are the same, then partial phase synchronization occurs. Moreover, if all natural frequencies are identical, we prove that all oscillators either achieve a complete phase synchronization asymptotically or tend to a bipolar phase-locking state. We also provide several numerical simulations to support the main results.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570758","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 3, Page 1705-1719, September 2024. Abstract.In 2006, Alexander proved a result that implies for a Weingarten surface [math], if [math] is the number of times a closed geodesic winds around the axis of rotation and [math] is the number of times the geodesic oscillates about the equator, then [math] when [math] and [math] when [math]. In this paper, we present another proof of Alexander’s result for the Weingarten surfaces [math] that is simpler and more direct. Our approach uses sharp estimates of certain improper integrals to obtain the intervals for permissible ratios [math]. We numerically compute a number of closed geodesics for various combinations of [math] to illustrate the variety of patterns that are possible.
{"title":"Closed Geodesics on Weingarten Surfaces with [math]","authors":"Frank E. Baginski, Valério Ramos Batista","doi":"10.1137/23m1608616","DOIUrl":"https://doi.org/10.1137/23m1608616","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1705-1719, September 2024. <br/> Abstract.In 2006, Alexander proved a result that implies for a Weingarten surface [math], if [math] is the number of times a closed geodesic winds around the axis of rotation and [math] is the number of times the geodesic oscillates about the equator, then [math] when [math] and [math] when [math]. In this paper, we present another proof of Alexander’s result for the Weingarten surfaces [math] that is simpler and more direct. Our approach uses sharp estimates of certain improper integrals to obtain the intervals for permissible ratios [math]. We numerically compute a number of closed geodesics for various combinations of [math] to illustrate the variety of patterns that are possible.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549742","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}
Pei Yu, Pantea Pooladvand, Mark M. Tanaka, Lindi M. Wahl
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1677-1703, June 2024. Abstract.Nearly all emerging diseases in humans are a result of host-range expansion, in which a pathogen of one species evolves the ability to infect a new host species. To present a rigorous analysis of pathogen host-range expansion, we derive a Lotka–Volterra dynamical system with two competing host species and a single parasite species; the parasite infects only one of the host species. We provide a stability and bifurcation analysis of this model. We then ask what happens if the parasite evolves the ability to infect the alternate host, extending the model to include a parasite population with an expanded host range. We derive explicit global stability and bifurcation conditions for this four-dimensional model in terms of the system parameters. We demonstrate that only four outcomes may occur following the range expansion of a parasite or pathogen, and provide both local and global asymptotic stability conditions for these outcomes. While three of these outcomes were expected, the fourth is counterintuitive, predicting that host-range expansion can drive the original host species to extinction. For example, a native species could be driven to extinction by a longstanding native parasite if that parasite acquires the ability to infect a cultivated species. We briefly discuss the phenomena driving this unexpected prediction and its implications.
{"title":"Extinctions Caused by Host-Range Expansion","authors":"Pei Yu, Pantea Pooladvand, Mark M. Tanaka, Lindi M. Wahl","doi":"10.1137/23m1605582","DOIUrl":"https://doi.org/10.1137/23m1605582","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1677-1703, June 2024. <br/> Abstract.Nearly all emerging diseases in humans are a result of host-range expansion, in which a pathogen of one species evolves the ability to infect a new host species. To present a rigorous analysis of pathogen host-range expansion, we derive a Lotka–Volterra dynamical system with two competing host species and a single parasite species; the parasite infects only one of the host species. We provide a stability and bifurcation analysis of this model. We then ask what happens if the parasite evolves the ability to infect the alternate host, extending the model to include a parasite population with an expanded host range. We derive explicit global stability and bifurcation conditions for this four-dimensional model in terms of the system parameters. We demonstrate that only four outcomes may occur following the range expansion of a parasite or pathogen, and provide both local and global asymptotic stability conditions for these outcomes. While three of these outcomes were expected, the fourth is counterintuitive, predicting that host-range expansion can drive the original host species to extinction. For example, a native species could be driven to extinction by a longstanding native parasite if that parasite acquires the ability to infect a cultivated species. We briefly discuss the phenomena driving this unexpected prediction and its implications.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505677","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 2, Page 1610-1635, June 2024. Abstract.The bifurcation from a simple eigenvalue (BSE) theorem is the foundation of steady-state bifurcation theory for one-parameter families of functions. When eigenvalues of multiplicity greater than one are caused by symmetry, the equivariant branching lemma (EBL) can often be applied to predict the branching of solutions. The EBL can be interpreted as the application of the BSE theorem to a fixed point subspace. There are functions which have invariant linear subspaces that are not caused by symmetry. For example, networks of identical coupled cells often have such invariant subspaces. We present a generalization of the EBL, where the BSE theorem is applied to nested invariant subspaces. We call this the bifurcation lemma for invariant subspaces (BLIS). We give several examples of bifurcations and determine if BSE, EBL, or BLIS applies. We extend our previous automated bifurcation analysis algorithms to use the BLIS to simplify and improve the detection of branches created at bifurcations.
{"title":"A Bifurcation Lemma for Invariant Subspaces","authors":"John M. Neuberger, Nándor Sieben, James W. Swift","doi":"10.1137/23m1595540","DOIUrl":"https://doi.org/10.1137/23m1595540","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1610-1635, June 2024. <br/> Abstract.The bifurcation from a simple eigenvalue (BSE) theorem is the foundation of steady-state bifurcation theory for one-parameter families of functions. When eigenvalues of multiplicity greater than one are caused by symmetry, the equivariant branching lemma (EBL) can often be applied to predict the branching of solutions. The EBL can be interpreted as the application of the BSE theorem to a fixed point subspace. There are functions which have invariant linear subspaces that are not caused by symmetry. For example, networks of identical coupled cells often have such invariant subspaces. We present a generalization of the EBL, where the BSE theorem is applied to nested invariant subspaces. We call this the bifurcation lemma for invariant subspaces (BLIS). We give several examples of bifurcations and determine if BSE, EBL, or BLIS applies. We extend our previous automated bifurcation analysis algorithms to use the BLIS to simplify and improve the detection of branches created at bifurcations.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505689","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 2, Page 1636-1676, June 2024. Abstract.Phase reduction is a well-established method to study weakly driven and weakly perturbed oscillators. Traditional phase-reduction approaches characterize the perturbed system dynamics solely in terms of the timing of the oscillations. In the case of large perturbations, the introduction of amplitude (isostable) coordinates improves the accuracy of the phase description by providing a sense of distance from the underlying limit cycle. Importantly, phase-amplitude coordinates allow for the study of both the timing and shape of system oscillations. A parallel tool is the infinitesimal shape response curve (iSRC), a variational method that characterizes the shape change of a limit-cycle oscillator under sustained perturbation. Despite the importance of oscillation amplitude in a wide range of physical systems, systematic studies on the shape change of oscillations remain scarce. Both phase-amplitude coordinates and the iSRC represent methods to analyze oscillation shape change, yet a relationship between the two has not been previously explored. In this work, we establish the iSRC and phase-amplitude coordinates as complementary tools to study oscillation amplitude. We extend existing iSRC theory and specify conditions under which a general class of systems can be analyzed by the joint iSRC phase-amplitude approach. We show that the iSRC takes on a dramatically simple form in phase-amplitude coordinates, and directly relate the phase and isostable response curves to the iSRC. We apply our theory to weakly perturbed single oscillators, and to study the synchronization and entrainment of coupled oscillators.
{"title":"On the Relation between Infinitesimal Shape Response Curves and Phase-Amplitude Reduction for Single and Coupled Limit-Cycle Oscillators","authors":"Max Kreider, Peter J. Thomas","doi":"10.1137/23m1575159","DOIUrl":"https://doi.org/10.1137/23m1575159","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1636-1676, June 2024. <br/> Abstract.Phase reduction is a well-established method to study weakly driven and weakly perturbed oscillators. Traditional phase-reduction approaches characterize the perturbed system dynamics solely in terms of the timing of the oscillations. In the case of large perturbations, the introduction of amplitude (isostable) coordinates improves the accuracy of the phase description by providing a sense of distance from the underlying limit cycle. Importantly, phase-amplitude coordinates allow for the study of both the timing and shape of system oscillations. A parallel tool is the infinitesimal shape response curve (iSRC), a variational method that characterizes the shape change of a limit-cycle oscillator under sustained perturbation. Despite the importance of oscillation amplitude in a wide range of physical systems, systematic studies on the shape change of oscillations remain scarce. Both phase-amplitude coordinates and the iSRC represent methods to analyze oscillation shape change, yet a relationship between the two has not been previously explored. In this work, we establish the iSRC and phase-amplitude coordinates as complementary tools to study oscillation amplitude. We extend existing iSRC theory and specify conditions under which a general class of systems can be analyzed by the joint iSRC phase-amplitude approach. We show that the iSRC takes on a dramatically simple form in phase-amplitude coordinates, and directly relate the phase and isostable response curves to the iSRC. We apply our theory to weakly perturbed single oscillators, and to study the synchronization and entrainment of coupled oscillators.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505679","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 2, Page 1579-1609, June 2024. Abstract.We calculate epidemic thresholds and investigate the dynamics of a disease in a networked metapopulation. To study the specific role of mobility levels, we utilize the SIR-network model and consider a range of network structures. For star-shaped networks where all nodes only connect to a center, we obtain the same epidemic threshold formulas as previously found for fully connected networks, considering all nodes with the same infection rate except one. We thus create a new terminology by saying that fully connected and star-shaped networks have the Standard Threshold Property. Next, we analyze cycle-shaped networks, which yield epidemic thresholds different from those obtained using star-shaped networks. We then analyze more general classes of networks by combining the star, cycle, and other structures, obtaining classes of networks with the Standard Threshold Property. We present some conjectures on even more flexible networks and complete our analysis by presenting simulations to explore the epidemic dynamics for the different structures.
{"title":"Epidemic Thresholds and Disease Dynamics in Metapopulations: The Role of Network Structure and Human Mobility","authors":"Haridas K. Das, Lucas M. Stolerman","doi":"10.1137/23m1579522","DOIUrl":"https://doi.org/10.1137/23m1579522","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1579-1609, June 2024. <br/> Abstract.We calculate epidemic thresholds and investigate the dynamics of a disease in a networked metapopulation. To study the specific role of mobility levels, we utilize the SIR-network model and consider a range of network structures. For star-shaped networks where all nodes only connect to a center, we obtain the same epidemic threshold formulas as previously found for fully connected networks, considering all nodes with the same infection rate except one. We thus create a new terminology by saying that fully connected and star-shaped networks have the Standard Threshold Property. Next, we analyze cycle-shaped networks, which yield epidemic thresholds different from those obtained using star-shaped networks. We then analyze more general classes of networks by combining the star, cycle, and other structures, obtaining classes of networks with the Standard Threshold Property. We present some conjectures on even more flexible networks and complete our analysis by presenting simulations to explore the epidemic dynamics for the different structures.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505680","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}
Hildeberto Jardón-Kojakhmetov, Christian Kuehn, Iacopo P. Longo
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1540-1578, June 2024. Abstract.We study synchronization for linearly coupled temporal networks of heterogeneous time-dependent nonlinear agents via the convergence of attracting trajectories of each node. The results are obtained by constructing and studying the stability of a suitable linear nonautonomous problem bounding the evolution of the synchronization errors. Both the case of the entire network and that of only a cluster are addressed, and the persistence of the obtained synchronization against perturbation is also discussed. Furthermore, a sufficient condition for the existence of attracting trajectories of each node is given. In all cases, the considered dependence on time requires only local integrability, which is a very mild regularity assumption. Moreover, our results mainly depend on the network structure and its properties and achieve synchronization up to a constant in finite time. Hence they are quite suitable for applications. The applicability of the results is showcased via several examples: coupled van der Pol/FitzHugh–Nagumo oscillators, weighted/signed opinion dynamics, and coupled Lorenz systems.
{"title":"Persistent Synchronization of Heterogeneous Networks with Time-Dependent Linear Diffusive Coupling","authors":"Hildeberto Jardón-Kojakhmetov, Christian Kuehn, Iacopo P. Longo","doi":"10.1137/23m1602024","DOIUrl":"https://doi.org/10.1137/23m1602024","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1540-1578, June 2024. <br/> Abstract.We study synchronization for linearly coupled temporal networks of heterogeneous time-dependent nonlinear agents via the convergence of attracting trajectories of each node. The results are obtained by constructing and studying the stability of a suitable linear nonautonomous problem bounding the evolution of the synchronization errors. Both the case of the entire network and that of only a cluster are addressed, and the persistence of the obtained synchronization against perturbation is also discussed. Furthermore, a sufficient condition for the existence of attracting trajectories of each node is given. In all cases, the considered dependence on time requires only local integrability, which is a very mild regularity assumption. Moreover, our results mainly depend on the network structure and its properties and achieve synchronization up to a constant in finite time. Hence they are quite suitable for applications. The applicability of the results is showcased via several examples: coupled van der Pol/FitzHugh–Nagumo oscillators, weighted/signed opinion dynamics, and coupled Lorenz systems.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505681","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 2, Page 1504-1539, June 2024. Abstract.The sparse identification of nonlinear dynamics (SINDy) algorithm can be applied to stochastic differential equations (SDEs) to estimate the drift and the diffusion function using data from a realization of the SDE. The SINDy algorithm requires sample data from each of these functions, which is typically estimated numerically from the data of the state. We analyze the performance of the previously proposed estimates for the drift and the diffusion function to give bounds on the error for finite data. However, since this algorithm only converges as both the sampling frequency and the length of trajectory go to infinity, obtaining approximations within a certain tolerance may be infeasible. To combat this, we develop estimates with higher orders of accuracy for use in the SINDy framework. For a given sampling frequency, these estimates give more accurate approximations of the drift and diffusion functions, making SINDy a far more feasible system identification method.
{"title":"On Higher Order Drift and Diffusion Estimates for Stochastic SINDy","authors":"Mathias Wanner, Igor Mezić","doi":"10.1137/23m1567011","DOIUrl":"https://doi.org/10.1137/23m1567011","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1504-1539, June 2024. <br/> Abstract.The sparse identification of nonlinear dynamics (SINDy) algorithm can be applied to stochastic differential equations (SDEs) to estimate the drift and the diffusion function using data from a realization of the SDE. The SINDy algorithm requires sample data from each of these functions, which is typically estimated numerically from the data of the state. We analyze the performance of the previously proposed estimates for the drift and the diffusion function to give bounds on the error for finite data. However, since this algorithm only converges as both the sampling frequency and the length of trajectory go to infinity, obtaining approximations within a certain tolerance may be infeasible. To combat this, we develop estimates with higher orders of accuracy for use in the SINDy framework. For a given sampling frequency, these estimates give more accurate approximations of the drift and diffusion functions, making SINDy a far more feasible system identification method.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505690","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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