{"title":"(boldsymbol{N})-Body Oscillator Interactions of Higher-Order Coupling Functions","authors":"Youngmin Park, D. Wilson","doi":"10.1137/23m1594182","DOIUrl":"https://doi.org/10.1137/23m1594182","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141349319","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":null,"pages":null},"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}
{"title":"Circularization in the Damped Kepler Problem","authors":"K. U. Kristiansen, R. Ortega","doi":"10.1137/23m1623720","DOIUrl":"https://doi.org/10.1137/23m1623720","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141361860","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":null,"pages":null},"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}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1345-1371, June 2024. Abstract.In a recent article by Guglielmi and Hairer [SIAM J. Appl. Dyn. Syst., 14 (2015), pp. 1454–1477], an analysis in the [math] limit was proposed of regularized discontinuous ODEs in codimension-2 switching domains; this was obtained by studying a certain 2-dimensional system describing the so-called hidden dynamics. In particular, the existence of a unique limit solution was not proved in all cases, a few of which were labeled as ambiguous, and it was not clear whether or not the ambiguity could be resolved. In this paper, we show that it cannot be resolved in general. A first contribution of this paper is an illustration of the dependence of the limit solution on the form of the switching function. Considering the parameter dependence in the ambiguous class of discontinuous systems, a second contribution is a bifurcation analysis, revealing a range of possible behaviors. Finally, we investigate the sensitivity of solutions in the transition from codimension-2 domains to codimension-3 when there is a limit cycle in the hidden dynamics.
{"title":"Nonuniqueness Phenomena in Discontinuous Dynamical Systems and Their Regularizations","authors":"Alessia Andò, Roderick Edwards, Nicola Guglielmi","doi":"10.1137/23m1587488","DOIUrl":"https://doi.org/10.1137/23m1587488","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1345-1371, June 2024. <br/> Abstract.In a recent article by Guglielmi and Hairer [SIAM J. Appl. Dyn. Syst., 14 (2015), pp. 1454–1477], an analysis in the [math] limit was proposed of regularized discontinuous ODEs in codimension-2 switching domains; this was obtained by studying a certain 2-dimensional system describing the so-called hidden dynamics. In particular, the existence of a unique limit solution was not proved in all cases, a few of which were labeled as ambiguous, and it was not clear whether or not the ambiguity could be resolved. In this paper, we show that it cannot be resolved in general. A first contribution of this paper is an illustration of the dependence of the limit solution on the form of the switching function. Considering the parameter dependence in the ambiguous class of discontinuous systems, a second contribution is a bifurcation analysis, revealing a range of possible behaviors. Finally, we investigate the sensitivity of solutions in the transition from codimension-2 domains to codimension-3 when there is a limit cycle in the hidden dynamics.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168382","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 1313-1344, June 2024. Abstract.This article investigates the nonlinear stability and dynamic bifurcation for the equations governing the Marangoni convection of two superimposed immiscible liquids subject to temperature gradient perpendicular to the plate. First, we obtain the critical value of the Marangoni number and verify the stability exchange principle by adopting a hybrid method that combines theoretical analysis and numerical calculations. Second, we employ the energy method, probing the nonlinear stability and establishing the nonlinear thresholds of the Marangoni number. Third, we apply the technique of center manifold reduction to reduce the corresponding infinite-dimensional model to finite-dimensional ODEs. According to the ODEs, we establish a dynamic bifurcation theorem with the transition number that determines the bifurcation type of the model. Finally, we determine the nondimensional transition number and present related temporal and flow patterns by performing careful numerical computation.
{"title":"On the Stability and Bifurcation of Marangoni Convection of Two Immiscible Liquids with a Nondeformable Interface","authors":"Chao Xing, Daozhi Han, Quan Wang","doi":"10.1137/23m1584174","DOIUrl":"https://doi.org/10.1137/23m1584174","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1313-1344, June 2024. <br/> Abstract.This article investigates the nonlinear stability and dynamic bifurcation for the equations governing the Marangoni convection of two superimposed immiscible liquids subject to temperature gradient perpendicular to the plate. First, we obtain the critical value of the Marangoni number and verify the stability exchange principle by adopting a hybrid method that combines theoretical analysis and numerical calculations. Second, we employ the energy method, probing the nonlinear stability and establishing the nonlinear thresholds of the Marangoni number. Third, we apply the technique of center manifold reduction to reduce the corresponding infinite-dimensional model to finite-dimensional ODEs. According to the ODEs, we establish a dynamic bifurcation theorem with the transition number that determines the bifurcation type of the model. Finally, we determine the nondimensional transition number and present related temporal and flow patterns by performing careful numerical computation.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141517555","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}
{"title":"Analysis of Equilibria and Connecting Orbits in a Nonlinear Viral Infection Model","authors":"Mengfeng Sun, Yijun Lou, Xinchu Fu","doi":"10.1137/23m1578115","DOIUrl":"https://doi.org/10.1137/23m1578115","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140968944","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 1242-1271, June 2024. Abstract.This paper studies the dynamic interplay between inherent agent dynamics and velocity alignment in the ensemble of Cucker–Smale (CS) flocks under both fixed and switching interaction topologies. For the fixed topology network modeled by a rooted digraph, we provide sufficient frameworks for exponential convergence of flocking provided that the Lipschitz constant of nonlinear dynamics is less than a given bound in terms of initial state and system parameters. Similar to the CS system free of inherent dynamics, our results exhibit threshold phenomena depending on the quantity of collectivity in of the rooted digraph. For the case with switching topologies being rooted in a sequence of time-blocks, we present similar sufficient frameworks leading to convergence of flocking in terms of initial state and system parameters. Our results hold for both continuous time and discrete time. Numerical simulations are performed to illustrate our theoretical results.
{"title":"Flocks with Nonlinear Inherent Dynamics under Fixed and Switching Digraphs","authors":"Jiu-Gang Dong","doi":"10.1137/23m1574270","DOIUrl":"https://doi.org/10.1137/23m1574270","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1242-1271, June 2024. <br/> Abstract.This paper studies the dynamic interplay between inherent agent dynamics and velocity alignment in the ensemble of Cucker–Smale (CS) flocks under both fixed and switching interaction topologies. For the fixed topology network modeled by a rooted digraph, we provide sufficient frameworks for exponential convergence of flocking provided that the Lipschitz constant of nonlinear dynamics is less than a given bound in terms of initial state and system parameters. Similar to the CS system free of inherent dynamics, our results exhibit threshold phenomena depending on the quantity of collectivity in of the rooted digraph. For the case with switching topologies being rooted in a sequence of time-blocks, we present similar sufficient frameworks leading to convergence of flocking in terms of initial state and system parameters. Our results hold for both continuous time and discrete time. Numerical simulations are performed to illustrate our theoretical results.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925810","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 1199-1241, June 2024. Abstract.In recent years, classical epidemic models, which assume stationary behavior of individuals, have been extended to include an adaptive heterogeneous response of the population to the current state of the epidemic. However, it is widely accepted that human behavior can exhibit history-dependence as a consequence of learned experiences. This history-dependence is similar to the hysteresis effects that have been well studied in control theory. To illustrate the importance of history-dependence for epidemic theory, we study the dynamics of a variant of the SIRS model where individuals exhibit lazy-switch responses to prevalence dynamics. The resulting model, which includes the Preisach hysteresis operator, possesses a continuum of endemic equilibrium states characterized by different proportions of susceptible, infected, and recovered populations. We discuss stability properties of the endemic equilibrium set and relate them to the degree of heterogeneity of the adaptive response. In particular, our results suggest that heterogeneity promotes the convergence of the epidemic trajectory to an equilibrium state. Heterogeneity can be achieved by selective intervention policies targeting specific population groups. On the other hand, heterogeneous responses can lead to a higher peak of infection during the epidemic and a higher prevalence at the endemic equilibrium after the epidemic. These results support the argument that public health responses during the emergence of a new disease have long-term consequences for subsequent management efforts. The main mathematical contribution of this work is a new method of global stability analysis, which uses a family of Lyapunov functions corresponding to different branches of the hysteresis operator. It is well known that instability can result from switching from one flow to another even though each flow is stable (if the flows have different Lyapunov functions). We provide sufficient conditions for the convergence of trajectories to the equilibrium set for switched systems with the Preisach hysteresis operator.
{"title":"Global Stability of SIR Model with Heterogeneous Transmission Rate Modeled by the Preisach Operator","authors":"Ruofei Guan, Jana Kopfová, Dmitrii Rachinskii","doi":"10.1137/22m154274x","DOIUrl":"https://doi.org/10.1137/22m154274x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1199-1241, June 2024. <br/> Abstract.In recent years, classical epidemic models, which assume stationary behavior of individuals, have been extended to include an adaptive heterogeneous response of the population to the current state of the epidemic. However, it is widely accepted that human behavior can exhibit history-dependence as a consequence of learned experiences. This history-dependence is similar to the hysteresis effects that have been well studied in control theory. To illustrate the importance of history-dependence for epidemic theory, we study the dynamics of a variant of the SIRS model where individuals exhibit lazy-switch responses to prevalence dynamics. The resulting model, which includes the Preisach hysteresis operator, possesses a continuum of endemic equilibrium states characterized by different proportions of susceptible, infected, and recovered populations. We discuss stability properties of the endemic equilibrium set and relate them to the degree of heterogeneity of the adaptive response. In particular, our results suggest that heterogeneity promotes the convergence of the epidemic trajectory to an equilibrium state. Heterogeneity can be achieved by selective intervention policies targeting specific population groups. On the other hand, heterogeneous responses can lead to a higher peak of infection during the epidemic and a higher prevalence at the endemic equilibrium after the epidemic. These results support the argument that public health responses during the emergence of a new disease have long-term consequences for subsequent management efforts. The main mathematical contribution of this work is a new method of global stability analysis, which uses a family of Lyapunov functions corresponding to different branches of the hysteresis operator. It is well known that instability can result from switching from one flow to another even though each flow is stable (if the flows have different Lyapunov functions). We provide sufficient conditions for the convergence of trajectories to the equilibrium set for switched systems with the Preisach hysteresis operator.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942287","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 1159-1198, June 2024. Abstract.In this work we study the splitting distance of a rapidly perturbed pendulum [math] with [math] a [math]-periodic function and [math]. Systems of this kind undergo exponentially small splitting, and, when [math], it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided [math]. Our study focuses on the case [math], and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as [math]. On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency [math] in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with [math] where, for most [math], the perturbation satisfies [math]. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton–Jacobi formalism. The leading exponentially small term appears at order [math], where [math] is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.
{"title":"Splitting of Separatrices for Rapid Degenerate Perturbations of the Classical Pendulum","authors":"Inmaculada Baldomá, Tere M-Seara, Román Moreno","doi":"10.1137/23m1550992","DOIUrl":"https://doi.org/10.1137/23m1550992","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1159-1198, June 2024. <br/> Abstract.In this work we study the splitting distance of a rapidly perturbed pendulum [math] with [math] a [math]-periodic function and [math]. Systems of this kind undergo exponentially small splitting, and, when [math], it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided [math]. Our study focuses on the case [math], and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as [math]. On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency [math] in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with [math] where, for most [math], the perturbation satisfies [math]. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton–Jacobi formalism. The leading exponentially small term appears at order [math], where [math] is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925923","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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