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":"25 1","pages":""},"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":"48 1","pages":""},"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":"9 1","pages":""},"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":"3 1","pages":""},"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":"29 1","pages":""},"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":"25 1","pages":""},"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":"4 1","pages":""},"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":"344 1","pages":""},"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}
Heather Z. Brooks, Philip S. Chodrow, Mason A. Porter
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1442-1470, June 2024. Abstract.We study a nonlinear bounded-confidence model (BCM) of continuous-time opinion dynamics on networks with both persuadable individuals and zealots. The model is parameterized by a nonnegative scalar [math], which controls the steepness of a smooth influence function. This influence function encodes the relative weights that individuals place on the opinions of other individuals. When [math], this influence function recovers Taylor’s averaging model; when [math], the influence function converges to that of a modified Hegselmann–Krause (HK) BCM. Unlike the classical HK model, however, our sigmoidal bounded-confidence model (SBCM) is smooth for any finite [math]. We show that the set of steady states of our SBCM is qualitatively similar to that of the Taylor model when [math] is small and that the set of steady states approaches a subset of the set of steady states of a modified HK model as [math]. For certain special graph topologies, we give analytical descriptions of important features of the space of steady states. A notable result is a closed-form relationship between graph topology and the stability of polarized states in a simple special case that models echo chambers in social networks. Because the influence function of our BCM is smooth, we are able to study it with linear stability analysis, which is difficult to employ with the usual discontinuous influence functions in BCMs.
{"title":"Emergence of Polarization in a Sigmoidal Bounded-Confidence Model of Opinion Dynamics","authors":"Heather Z. Brooks, Philip S. Chodrow, Mason A. Porter","doi":"10.1137/22m1527258","DOIUrl":"https://doi.org/10.1137/22m1527258","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1442-1470, June 2024. <br/>Abstract.We study a nonlinear bounded-confidence model (BCM) of continuous-time opinion dynamics on networks with both persuadable individuals and zealots. The model is parameterized by a nonnegative scalar [math], which controls the steepness of a smooth influence function. This influence function encodes the relative weights that individuals place on the opinions of other individuals. When [math], this influence function recovers Taylor’s averaging model; when [math], the influence function converges to that of a modified Hegselmann–Krause (HK) BCM. Unlike the classical HK model, however, our sigmoidal bounded-confidence model (SBCM) is smooth for any finite [math]. We show that the set of steady states of our SBCM is qualitatively similar to that of the Taylor model when [math] is small and that the set of steady states approaches a subset of the set of steady states of a modified HK model as [math]. For certain special graph topologies, we give analytical descriptions of important features of the space of steady states. A notable result is a closed-form relationship between graph topology and the stability of polarized states in a simple special case that models echo chambers in social networks. Because the influence function of our BCM is smooth, we are able to study it with linear stability analysis, which is difficult to employ with the usual discontinuous influence functions in BCMs.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"31 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529306","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}
Sebin Gracy, Mengbin Ye, Brian D. O. Anderson, Cesar A. Uribe
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1372-1410, June 2024. Abstract.This paper studies the endemic behavior of a multi-competitive networked susceptible-infected-susceptible (SIS) model. Specifically, the paper deals with three competing virus systems (i.e., tri-virus systems) spreading over a population. First, we show that a tri-virus system, unlike a bi-virus system, is not a monotone dynamical system. Using the Parametric Transversality Theorem, we show that, generically, a tri-virus system has a finite number of equilibria and that the Jacobian matrices associated with each equilibrium are nonsingular. The endemic equilibria of this system can be classified as follows: (a) single-virus endemic equilibria (also referred to as the boundary equilibria), where precisely one of the three viruses is present in the population; (b) 2-coexistence equilibria, where exactly two of the three viruses are present in the population; and (c) 3-coexistence equilibria, where all three viruses present in the population. By leveraging the notions of basic reproduction number (i.e., the number of infections caused by an infected individual in a completely susceptible population) and invasion reproduction number (i.e., the average number of infections caused by an individual in a setting where other endemic virus(es) are at equilibrium), we provide a necessary and sufficient condition that guarantees local exponential convergence to a boundary equilibrium. Further, we secure conditions for the nonexistence of 3-coexistence equilibria (resp., for various kinds of 2-coexistence equilibria). We also identify sufficient conditions for the existence of a 2-coexistence (resp., 3-coexistence) equilibrium. We identify conditions on the model parameters that give rise to a continuum of coexistence equilibria. More specifically, we establish (i) a scenario that admits the existence and local exponential attractivity of a line of coexistence equilibria; and (ii) scenarios that admit the existence of, and, in the case of one such scenario, global convergence to, a plane of 3-coexistence equilibria.
{"title":"Towards Understanding the Endemic Behavior of a Competitive Tri-virus SIS Networked Model","authors":"Sebin Gracy, Mengbin Ye, Brian D. O. Anderson, Cesar A. Uribe","doi":"10.1137/23m1563074","DOIUrl":"https://doi.org/10.1137/23m1563074","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1372-1410, June 2024. <br/> Abstract.This paper studies the endemic behavior of a multi-competitive networked susceptible-infected-susceptible (SIS) model. Specifically, the paper deals with three competing virus systems (i.e., tri-virus systems) spreading over a population. First, we show that a tri-virus system, unlike a bi-virus system, is not a monotone dynamical system. Using the Parametric Transversality Theorem, we show that, generically, a tri-virus system has a finite number of equilibria and that the Jacobian matrices associated with each equilibrium are nonsingular. The endemic equilibria of this system can be classified as follows: (a) single-virus endemic equilibria (also referred to as the boundary equilibria), where precisely one of the three viruses is present in the population; (b) 2-coexistence equilibria, where exactly two of the three viruses are present in the population; and (c) 3-coexistence equilibria, where all three viruses present in the population. By leveraging the notions of basic reproduction number (i.e., the number of infections caused by an infected individual in a completely susceptible population) and invasion reproduction number (i.e., the average number of infections caused by an individual in a setting where other endemic virus(es) are at equilibrium), we provide a necessary and sufficient condition that guarantees local exponential convergence to a boundary equilibrium. Further, we secure conditions for the nonexistence of 3-coexistence equilibria (resp., for various kinds of 2-coexistence equilibria). We also identify sufficient conditions for the existence of a 2-coexistence (resp., 3-coexistence) equilibrium. We identify conditions on the model parameters that give rise to a continuum of coexistence equilibria. More specifically, we establish (i) a scenario that admits the existence and local exponential attractivity of a line of coexistence equilibria; and (ii) scenarios that admit the existence of, and, in the case of one such scenario, global convergence to, a plane of 3-coexistence equilibria.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"9 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505717","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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