Bogdan Batko, Marcio Gameiro, Ying Hung, William Kalies, Konstantin Mischaikow, Ewerton Vieira
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 383-409, March 2024. Abstract.We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system (assumed to be a sample path of the GP). The focus of this paper is on explaining the ideas, thus we restrict our examples to one-dimensional systems and show how to capture the existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics.
{"title":"Identifying Nonlinear Dynamics with High Confidence from Sparse Data","authors":"Bogdan Batko, Marcio Gameiro, Ying Hung, William Kalies, Konstantin Mischaikow, Ewerton Vieira","doi":"10.1137/23m1560252","DOIUrl":"https://doi.org/10.1137/23m1560252","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 383-409, March 2024. <br/> Abstract.We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system (assumed to be a sample path of the GP). The focus of this paper is on explaining the ideas, thus we restrict our examples to one-dimensional systems and show how to capture the existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 358-382, March 2024. Abstract. This paper is mainly concerned with limiting behaviors of invariant measures for neural field lattice models in a random environment. First of all, we consider the convergence relation of invariant measures between the stochastic neural field lattice model and the corresponding deterministic model in weighted spaces, and prove any limit of a sequence of invariant measures of such a lattice model must be an invariant measure of its limiting system as the noise intensity tends to zero. Then we are devoted to studying the numerical approximation of invariant measures of such a stochastic neural lattice model. To this end, we first consider convergence of invariant measures between such a neural lattice model and the system with neurons only interacting with its n-neighborhood; then we further prove the convergence relation of invariant measures between the system with an n-neighborhood and its finite dimensional truncated system. By this procedure, the invariant measure of the stochastic neural lattice models can be approximated by the numerical invariant measure of a finite dimensional truncated system based on the backward Euler–Maruyama (BEM) scheme. Therefore, the invariant measure of a deterministic neural field lattice model can be observed by the invariant measure of the BEM scheme when the noise is not negligible.
SIAM 应用动力系统期刊》，第 23 卷第 1 期，第 358-382 页，2024 年 3 月。 摘要本文主要研究随机环境下神经场晶格模型不变度量的极限行为。首先，我们考虑了随机神经场网格模型与相应的确定性模型在加权空间中的不变量度量收敛关系，并证明了当噪声强度趋于零时，该网格模型不变量度量序列的任何极限必定是其极限系统的不变量度量。然后，我们将致力于研究这种随机神经网格模型不变量的数值逼近。为此，我们首先考虑这种神经网格模型与神经元只与其 n 邻域相互作用的系统之间的不变量度量的收敛性；然后，我们进一步证明具有 n 邻域的系统与其有限维截断系统之间的不变量度量的收敛关系。通过这一过程，随机神经网格模型的不变度量可以用基于后向欧拉-马鲁山（BEM）方案的有限维截断系统的数值不变度量来近似。因此，当噪声不可忽略时，确定性神经场网格模型的不变度量可以通过 BEM 方案的不变度量来观察。
{"title":"Convergence and Approximation of Invariant Measures for Neural Field Lattice Models under Noise Perturbation","authors":"Tomas Caraballo, Zhang Chen, Lingyu Li","doi":"10.1137/23m157137x","DOIUrl":"https://doi.org/10.1137/23m157137x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 358-382, March 2024. <br/> Abstract. This paper is mainly concerned with limiting behaviors of invariant measures for neural field lattice models in a random environment. First of all, we consider the convergence relation of invariant measures between the stochastic neural field lattice model and the corresponding deterministic model in weighted spaces, and prove any limit of a sequence of invariant measures of such a lattice model must be an invariant measure of its limiting system as the noise intensity tends to zero. Then we are devoted to studying the numerical approximation of invariant measures of such a stochastic neural lattice model. To this end, we first consider convergence of invariant measures between such a neural lattice model and the system with neurons only interacting with its n-neighborhood; then we further prove the convergence relation of invariant measures between the system with an n-neighborhood and its finite dimensional truncated system. By this procedure, the invariant measure of the stochastic neural lattice models can be approximated by the numerical invariant measure of a finite dimensional truncated system based on the backward Euler–Maruyama (BEM) scheme. Therefore, the invariant measure of a deterministic neural field lattice model can be observed by the invariant measure of the BEM scheme when the noise is not negligible.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554595","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}
Olivia Cannon, Ty Bondurant, Malindi Whyte, Arnd Scheel
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 297-324, March 2024. Abstract. We investigate the effect of bias on the formation and dynamics of opinion clusters in the bounded confidence model. For weak bias, we quantify the change in average opinion and potential dispersion and decrease in cluster size. For nonlinear bias modeling self-incitement, we establish coherent drifting motion of clusters on a background of uniform opinion distribution for biases below a critical threshold where clusters dissolve. Technically, we use geometric singular perturbation theory to derive drift speeds, we rely on a nonlocal center manifold analysis to construct drifting clusters near threshold, and we implement numerical continuation in a forward-backward delay equation to connect asymptotic regimes.
{"title":"Shifting Consensus in a Biased Compromise Model","authors":"Olivia Cannon, Ty Bondurant, Malindi Whyte, Arnd Scheel","doi":"10.1137/23m1552346","DOIUrl":"https://doi.org/10.1137/23m1552346","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 297-324, March 2024. <br/> Abstract. We investigate the effect of bias on the formation and dynamics of opinion clusters in the bounded confidence model. For weak bias, we quantify the change in average opinion and potential dispersion and decrease in cluster size. For nonlinear bias modeling self-incitement, we establish coherent drifting motion of clusters on a background of uniform opinion distribution for biases below a critical threshold where clusters dissolve. Technically, we use geometric singular perturbation theory to derive drift speeds, we rely on a nonlocal center manifold analysis to construct drifting clusters near threshold, and we implement numerical continuation in a forward-backward delay equation to connect asymptotic regimes.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"3 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 325-357, March 2024. Abstract. Generalized mass-action systems are power-law dynamical systems arising from chemical reaction networks. Essentially, every nonnegative ODE model used in chemistry and biology (for example, in ecology and epidemiology) and even in economics and engineering can be written in this form. Previous results have focused on existence and uniqueness of special steady states (complex-balanced equilibria) for all rate constants, thereby ruling out multiple (special) steady states. Recently, necessary conditions for linear stability have been obtained. In this work, we provide sufficient conditions for the linear stability of complex-balanced equilibria for all rate constants (and also for the nonexistence of other steady states). In particular, via sign vector conditions (on the stoichiometric coefficients and kinetic orders), we guarantee that the Jacobian matrix is a [math]-matrix. Technically, we use a new decomposition of the graph Laplacian which allows us to consider orders of (generalized) monomials. Alternatively, we use cycle decomposition which allows a linear parametrization of all Jacobian matrices. In any case, we guarantee stability without explicit computation of steady states. We illustrate our results in examples from chemistry and biology: generalized Lotka–Volterra systems and SIR models, a two-component signaling system, and an enzymatic futile cycle.
{"title":"Sufficient Conditions for Linear Stability of Complex-Balanced Equilibria in Generalized Mass-Action Systems","authors":"Stefan Müller, Georg Regensburger","doi":"10.1137/22m154260x","DOIUrl":"https://doi.org/10.1137/22m154260x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 325-357, March 2024. <br/> Abstract. Generalized mass-action systems are power-law dynamical systems arising from chemical reaction networks. Essentially, every nonnegative ODE model used in chemistry and biology (for example, in ecology and epidemiology) and even in economics and engineering can be written in this form. Previous results have focused on existence and uniqueness of special steady states (complex-balanced equilibria) for all rate constants, thereby ruling out multiple (special) steady states. Recently, necessary conditions for linear stability have been obtained. In this work, we provide sufficient conditions for the linear stability of complex-balanced equilibria for all rate constants (and also for the nonexistence of other steady states). In particular, via sign vector conditions (on the stoichiometric coefficients and kinetic orders), we guarantee that the Jacobian matrix is a [math]-matrix. Technically, we use a new decomposition of the graph Laplacian which allows us to consider orders of (generalized) monomials. Alternatively, we use cycle decomposition which allows a linear parametrization of all Jacobian matrices. In any case, we guarantee stability without explicit computation of steady states. We illustrate our results in examples from chemistry and biology: generalized Lotka–Volterra systems and SIR models, a two-component signaling system, and an enzymatic futile cycle.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"48 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554630","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}
Aurélien Desoeuvres, Alexandru Iosif, Christoph Lüders, Ovidiu Radulescu, Hamid Rahkooy, Matthias Seiß, Thomas Sturm
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 256-296, March 2024. Abstract. Model reduction of fast-slow chemical reaction networks based on the quasi-steady state approximation fails when the fast subsystem has first integrals. We call these first integrals approximate conservation laws. In order to define fast subsystems and identify approximate conservation laws, we use ideas from tropical geometry. We prove that any approximate conservation law evolves more slowly than all the species involved in it and therefore represents a supplementary slow variable in an extended system. By elimination of some variables of the extended system, we obtain networks without approximate conservation laws, which can be reduced by standard singular perturbation methods. The field of applications of approximate conservation laws covers the quasi-equilibrium approximation, which is well known in biochemistry. We discuss reductions of slow-fast as well as multiple timescale systems. Networks with multiple timescales have hierarchical relaxation. At a given timescale, our multiple timescale reduction method defines three subsystems composed of (i) slaved fast variables satisfying algebraic equations, (ii) slow driving variables satisfying reduced ordinary differential equations, and (iii) quenched much slower variables that are constant. The algebraic equations satisfied by fast variables define chains of nested normally hyperbolic invariant manifolds. In such chains, faster manifolds are of higher dimension and contain the slower manifolds. Our reduction methods are introduced algorithmically for networks with monomial reaction rates and linear, monomial, or polynomial approximate conservation laws. We propose symbolic algorithms to reshape and rescale the networks such that geometric singular perturbation theory can be applied to them, test the applicability of the theory, and finally reduce the networks. As a proof of concept, we apply this method to a model of the TGF-beta signaling pathway.
{"title":"Reduction of Chemical Reaction Networks with Approximate Conservation Laws","authors":"Aurélien Desoeuvres, Alexandru Iosif, Christoph Lüders, Ovidiu Radulescu, Hamid Rahkooy, Matthias Seiß, Thomas Sturm","doi":"10.1137/22m1543963","DOIUrl":"https://doi.org/10.1137/22m1543963","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 256-296, March 2024. <br/> Abstract. Model reduction of fast-slow chemical reaction networks based on the quasi-steady state approximation fails when the fast subsystem has first integrals. We call these first integrals approximate conservation laws. In order to define fast subsystems and identify approximate conservation laws, we use ideas from tropical geometry. We prove that any approximate conservation law evolves more slowly than all the species involved in it and therefore represents a supplementary slow variable in an extended system. By elimination of some variables of the extended system, we obtain networks without approximate conservation laws, which can be reduced by standard singular perturbation methods. The field of applications of approximate conservation laws covers the quasi-equilibrium approximation, which is well known in biochemistry. We discuss reductions of slow-fast as well as multiple timescale systems. Networks with multiple timescales have hierarchical relaxation. At a given timescale, our multiple timescale reduction method defines three subsystems composed of (i) slaved fast variables satisfying algebraic equations, (ii) slow driving variables satisfying reduced ordinary differential equations, and (iii) quenched much slower variables that are constant. The algebraic equations satisfied by fast variables define chains of nested normally hyperbolic invariant manifolds. In such chains, faster manifolds are of higher dimension and contain the slower manifolds. Our reduction methods are introduced algorithmically for networks with monomial reaction rates and linear, monomial, or polynomial approximate conservation laws. We propose symbolic algorithms to reshape and rescale the networks such that geometric singular perturbation theory can be applied to them, test the applicability of the theory, and finally reduce the networks. As a proof of concept, we apply this method to a model of the TGF-beta signaling pathway.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139497403","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}
Jacob Stærk-Østergaard, Anders Rahbek, Susanne Ditlevsen
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 236-255, March 2024. Abstract. This paper presents a novel estimator for a nonstandard restriction to both symmetry and low rank in the context of high-dimensional cointegrated processes. Furthermore, we discuss rank estimation for high-dimensional cointegrated processes by restricted bootstrapping of the Gaussian innovations. We demonstrate that the classical rank test for cointegrated systems is prone to underestimating the true rank and demonstrate this effect in a 100-dimensional system. We also discuss the implications of this underestimation for such high-dimensional systems in general. Also, we define a linearized Kuramoto system and present a simulation study, where we infer the cointegration rank of the unrestricted [math] system and successively the underlying clustered network structure based on a graphical approach and a symmetrized low rank estimator of the couplings derived from a reparametrization of the likelihood under this unusual restriction.
{"title":"High-Dimensional Cointegration and Kuramoto Inspired Systems","authors":"Jacob Stærk-Østergaard, Anders Rahbek, Susanne Ditlevsen","doi":"10.1137/22m1509771","DOIUrl":"https://doi.org/10.1137/22m1509771","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 236-255, March 2024. <br/> Abstract. This paper presents a novel estimator for a nonstandard restriction to both symmetry and low rank in the context of high-dimensional cointegrated processes. Furthermore, we discuss rank estimation for high-dimensional cointegrated processes by restricted bootstrapping of the Gaussian innovations. We demonstrate that the classical rank test for cointegrated systems is prone to underestimating the true rank and demonstrate this effect in a 100-dimensional system. We also discuss the implications of this underestimation for such high-dimensional systems in general. Also, we define a linearized Kuramoto system and present a simulation study, where we infer the cointegration rank of the unrestricted [math] system and successively the underlying clustered network structure based on a graphical approach and a symmetrized low rank estimator of the couplings derived from a reparametrization of the likelihood under this unusual restriction.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"8 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139497303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 205-235, March 2024. Abstract.In view of the molecular biological mechanism of the cytotoxic T lymphocytes proliferation induced by hepatitis B virus infection in vivo, a novel dynamical model with interval delay is proposed. The interval delay is determined by two delay parameters, namely delay center and delay radius. We derive the basic reproduction number [math] for the viral infection and obtain that the virus-free equilibrium (VFE) is globally asymptotically stable if [math]. When [math], besides VFE, the unique virus-present equilibrium (VPE) exists and the conditions of its asymptotical stability are obtained. Moreover, we study the Hopf bifurcations induced by the two delay parameters. Although there is no mitotic term in the target-cell dynamics, the results indicate that both these delay parameters can lead to periodic fluctuations at VPE, but only the smaller delay radius will destabilize the system, which is different from the classical discrete delay or distributed delay. Numerical simulations indicate that the proposed model can capture the profiles of the clinical data of two untreated chronic hepatitis B patients. The ability of interval delay to destabilize the system is between discrete delay and distributed delay, and the delay center plays the primary role. Pharmaceutical treatment can affect the stability of VPE and induce the fast-slow periodic phenomenon.
{"title":"Dynamics on Hepatitis B Virus Infection In Vivo with Interval Delay","authors":"Haonan Zhong, Kaifa Wang","doi":"10.1137/23m154546x","DOIUrl":"https://doi.org/10.1137/23m154546x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 205-235, March 2024. <br/> Abstract.In view of the molecular biological mechanism of the cytotoxic T lymphocytes proliferation induced by hepatitis B virus infection in vivo, a novel dynamical model with interval delay is proposed. The interval delay is determined by two delay parameters, namely delay center and delay radius. We derive the basic reproduction number [math] for the viral infection and obtain that the virus-free equilibrium (VFE) is globally asymptotically stable if [math]. When [math], besides VFE, the unique virus-present equilibrium (VPE) exists and the conditions of its asymptotical stability are obtained. Moreover, we study the Hopf bifurcations induced by the two delay parameters. Although there is no mitotic term in the target-cell dynamics, the results indicate that both these delay parameters can lead to periodic fluctuations at VPE, but only the smaller delay radius will destabilize the system, which is different from the classical discrete delay or distributed delay. Numerical simulations indicate that the proposed model can capture the profiles of the clinical data of two untreated chronic hepatitis B patients. The ability of interval delay to destabilize the system is between discrete delay and distributed delay, and the delay center plays the primary role. Pharmaceutical treatment can affect the stability of VPE and induce the fast-slow periodic phenomenon.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139497360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 167-204, March 2024. Abstract.We analyze the dynamics of networks in which a central pattern generator (CPG) transmits signals along one or more feedforward chains in a synchronous or phase-synchronous manner. Such propagating signals are common in biology, especially in locomotion and peristalsis, and are of interest for continuum robots. We construct such networks as feedforward lifts of the CPG. If the CPG dynamics is periodic, so is the lifted dynamics. Synchrony with the CPG manifests as a standing wave, and a regular phase pattern creates a traveling wave. We discuss Liapunov, asymptotic, and Floquet stability of the lifted periodic orbit and introduce transverse versions of these conditions that imply stability for signals propagating along arbitrarily long chains. We compare these notions to a simpler condition, transverse stability of the synchrony subspace, which is equivalent to Floquet stability when nodes are 1 dimensional.
{"title":"Stable Synchronous Propagation of Signals by Feedforward Networks","authors":"Ian Stewart, David Wood","doi":"10.1137/23m1552267","DOIUrl":"https://doi.org/10.1137/23m1552267","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 167-204, March 2024. <br/> Abstract.We analyze the dynamics of networks in which a central pattern generator (CPG) transmits signals along one or more feedforward chains in a synchronous or phase-synchronous manner. Such propagating signals are common in biology, especially in locomotion and peristalsis, and are of interest for continuum robots. We construct such networks as feedforward lifts of the CPG. If the CPG dynamics is periodic, so is the lifted dynamics. Synchrony with the CPG manifests as a standing wave, and a regular phase pattern creates a traveling wave. We discuss Liapunov, asymptotic, and Floquet stability of the lifted periodic orbit and introduce transverse versions of these conditions that imply stability for signals propagating along arbitrarily long chains. We compare these notions to a simpler condition, transverse stability of the synchrony subspace, which is equivalent to Floquet stability when nodes are 1 dimensional.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139482553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 127-166, March 2024. Abstract. The aim of this paper is to present a method to compute parameterizations of partially hyperbolic invariant tori and their invariant bundles in nonautonomous quasi-periodic Hamiltonian systems. We generalize flow map parameterization methods to the quasi-periodic setting. To this end, we introduce the notion of fiberwise isotropic tori and sketch definitions and results on fiberwise symplectic deformations and their moment maps. These constructs are vital to work in a suitable setting and lead to the proofs of “magic cancellations” that guarantee the existence of solutions of cohomological equations. We apply our algorithms in the elliptic restricted three body problem and compute nonresonant 3-dimensional invariant tori and their invariant bundles around the [math] point.
{"title":"Flow Map Parameterization Methods for Invariant Tori in Quasi-Periodic Hamiltonian Systems","authors":"Álvaro Fernández-Mora, Alex Haro, J. M. Mondelo","doi":"10.1137/23m1561257","DOIUrl":"https://doi.org/10.1137/23m1561257","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 127-166, March 2024. <br/> Abstract. The aim of this paper is to present a method to compute parameterizations of partially hyperbolic invariant tori and their invariant bundles in nonautonomous quasi-periodic Hamiltonian systems. We generalize flow map parameterization methods to the quasi-periodic setting. To this end, we introduce the notion of fiberwise isotropic tori and sketch definitions and results on fiberwise symplectic deformations and their moment maps. These constructs are vital to work in a suitable setting and lead to the proofs of “magic cancellations” that guarantee the existence of solutions of cohomological equations. We apply our algorithms in the elliptic restricted three body problem and compute nonresonant 3-dimensional invariant tori and their invariant bundles around the [math] point.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"14 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139464007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 98-126, March 2024. Abstract. This paper presents a methodology for the computation of whole sets of heteroclinic connections between isoenergetic slices of center manifolds of center [math] center [math] saddle fixed points of autonomous Hamiltonian systems. It involves (a) computing Taylor expansions of the center-unstable and center-stable manifolds of the departing and arriving fixed points through the parameterization method, using a new style that uncouples the center part from the hyperbolic one, thus making the fibered structure of the manifolds explicit; (b) uniformly meshing isoenergetic slices of the center manifolds, using a novel strategy that avoids numerical integration of the reduced differential equations and makes an explicit three-dimensional representation of these slices as deformed solid ellipsoids; (c) matching the center-stable and center-unstable manifolds of the departing and arriving points in a Poincaré section. The methodology is applied to obtain the whole set of isoenergetic heteroclinic connections from the center manifold of [math] to the center manifold of [math] in the Earth-Moon circular, spatial restricted three-body problem, for nine increasing energy levels that reach the appearance of halo orbits in both [math] and [math]. Some comments are made on possible applications to space mission design.
{"title":"Semianalytical Computation of Heteroclinic Connections Between Center Manifolds with the Parameterization Method","authors":"Miquel Barcelona, Alex Haro, Josep-Maria Mondelo","doi":"10.1137/23m1547883","DOIUrl":"https://doi.org/10.1137/23m1547883","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 98-126, March 2024. <br/> Abstract. This paper presents a methodology for the computation of whole sets of heteroclinic connections between isoenergetic slices of center manifolds of center [math] center [math] saddle fixed points of autonomous Hamiltonian systems. It involves (a) computing Taylor expansions of the center-unstable and center-stable manifolds of the departing and arriving fixed points through the parameterization method, using a new style that uncouples the center part from the hyperbolic one, thus making the fibered structure of the manifolds explicit; (b) uniformly meshing isoenergetic slices of the center manifolds, using a novel strategy that avoids numerical integration of the reduced differential equations and makes an explicit three-dimensional representation of these slices as deformed solid ellipsoids; (c) matching the center-stable and center-unstable manifolds of the departing and arriving points in a Poincaré section. The methodology is applied to obtain the whole set of isoenergetic heteroclinic connections from the center manifold of [math] to the center manifold of [math] in the Earth-Moon circular, spatial restricted three-body problem, for nine increasing energy levels that reach the appearance of halo orbits in both [math] and [math]. Some comments are made on possible applications to space mission design.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"48 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102944","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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