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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
Tamal K. Dey, Michał Lipiński, Marian Mrozek, Ryan Slechta
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 81-97, March 2024. Abstract. Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the context of a rapidly growing data science that requires new mathematical tools for discrete data. Toward this goal, the classical theory for connection matrices has been adapted to combinatorial frameworks that facilitate computation. We develop an efficient persistence-like algorithm to compute a connection matrix from a given combinatorial (multi) vector field on a simplicial complex. This algorithm requires a single pass, improving upon a known algorithm that runs an implicit recursion executing two passes at each level. Overall, the new algorithm is more simple, direct, and efficient than the state-of-the-art. Because of the algorithm’s similarity to the persistence algorithm, one may take advantage of various software optimizations from topological data analysis.
{"title":"Computing Connection Matrices via Persistence-Like Reductions","authors":"Tamal K. Dey, Michał Lipiński, Marian Mrozek, Ryan Slechta","doi":"10.1137/23m1562469","DOIUrl":"https://doi.org/10.1137/23m1562469","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 81-97, March 2024. <br/> Abstract. Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the context of a rapidly growing data science that requires new mathematical tools for discrete data. Toward this goal, the classical theory for connection matrices has been adapted to combinatorial frameworks that facilitate computation. We develop an efficient persistence-like algorithm to compute a connection matrix from a given combinatorial (multi) vector field on a simplicial complex. This algorithm requires a single pass, improving upon a known algorithm that runs an implicit recursion executing two passes at each level. Overall, the new algorithm is more simple, direct, and efficient than the state-of-the-art. Because of the algorithm’s similarity to the persistence algorithm, one may take advantage of various software optimizations from topological data analysis.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102951","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 50-80, March 2024. Abstract. Two competing types of interactions often play an important part in shaping system behavior, such as activatory and inhibitory functions in biological systems. Hence, signed networks, where each connection can be either positive or negative, have become popular models over recent years. However, the primary focus of the literature is on the unweighted and structurally balanced ones, where all cycles have an even number of negative edges. Hence here, we first introduce a classification of signed networks into balanced, antibalanced, or strictly unbalanced ones, and then characterize each type of signed networks in terms of the spectral properties of the signed weighted adjacency matrix. In particular, we show that the spectral radius of the matrix with signs is smaller than that without if and only if the signed network is strictly unbalanced. These properties are important to understand the dynamics on signed networks, both linear and nonlinear ones. Specifically, we find consistent patterns in a linear and a nonlinear dynamics theoretically, depending on their type of balance. We also propose two measures to further characterize strictly unbalanced networks, motivated by perturbation theory. Finally, we numerically verify these properties through experiments on both synthetic and real networks.
{"title":"Spreading and Structural Balance on Signed Networks","authors":"Yu Tian, Renaud Lambiotte","doi":"10.1137/22m1542325","DOIUrl":"https://doi.org/10.1137/22m1542325","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 50-80, March 2024. <br/> Abstract. Two competing types of interactions often play an important part in shaping system behavior, such as activatory and inhibitory functions in biological systems. Hence, signed networks, where each connection can be either positive or negative, have become popular models over recent years. However, the primary focus of the literature is on the unweighted and structurally balanced ones, where all cycles have an even number of negative edges. Hence here, we first introduce a classification of signed networks into balanced, antibalanced, or strictly unbalanced ones, and then characterize each type of signed networks in terms of the spectral properties of the signed weighted adjacency matrix. In particular, we show that the spectral radius of the matrix with signs is smaller than that without if and only if the signed network is strictly unbalanced. These properties are important to understand the dynamics on signed networks, both linear and nonlinear ones. Specifically, we find consistent patterns in a linear and a nonlinear dynamics theoretically, depending on their type of balance. We also propose two measures to further characterize strictly unbalanced networks, motivated by perturbation theory. Finally, we numerically verify these properties through experiments on both synthetic and real networks.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102950","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}
L. I. Allen, T. G. Molnár, Z. Dombóvári, S. J. Hogan
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 1-25, March 2024. Abstract. In this paper, we analyze the celebrated Haken–Kelso–Bunz model, describing the dynamics of bimanual coordination, in the presence of delay. We study the linear dynamics, stability, nonlinear behavior, and bifurcations of this model by both theoretical and numerical analysis. We calculate in-phase and antiphase limit cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis and center manifold reduction. Moreover, we uncover further details on the global dynamic behavior by numerical continuation, including the occurrence of limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions.
{"title":"The Effects of Delay on the HKB Model of Human Motor Coordination","authors":"L. I. Allen, T. G. Molnár, Z. Dombóvári, S. J. Hogan","doi":"10.1137/22m1531518","DOIUrl":"https://doi.org/10.1137/22m1531518","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 1-25, March 2024. <br/> Abstract. In this paper, we analyze the celebrated Haken–Kelso–Bunz model, describing the dynamics of bimanual coordination, in the presence of delay. We study the linear dynamics, stability, nonlinear behavior, and bifurcations of this model by both theoretical and numerical analysis. We calculate in-phase and antiphase limit cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis and center manifold reduction. Moreover, we uncover further details on the global dynamic behavior by numerical continuation, including the occurrence of limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139093876","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 26-49, March 2024. Abstract. Contact defects are time-periodic patterns in one space dimension that resemble spatially homogeneous oscillations with a defect embedded in their core region. For theoretical and numerical purposes, it is important to understand whether these defects persist when the domain is truncated to large spatial intervals, supplemented by appropriate boundary conditions. The present work shows that truncated contact defects exist and are unique on sufficiently large spatial intervals.
{"title":"Truncation of Contact Defects in Reaction-Diffusion Systems","authors":"Milen Ivanov, Björn Sandstede","doi":"10.1137/23m1546257","DOIUrl":"https://doi.org/10.1137/23m1546257","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 26-49, March 2024. <br/> Abstract. Contact defects are time-periodic patterns in one space dimension that resemble spatially homogeneous oscillations with a defect embedded in their core region. For theoretical and numerical purposes, it is important to understand whether these defects persist when the domain is truncated to large spatial intervals, supplemented by appropriate boundary conditions. The present work shows that truncated contact defects exist and are unique on sufficiently large spatial intervals.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139093581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3390-3430, December 2023. Abstract. For many engineered systems it is important to assess vulnerability to potential disturbances in order to ensure reliable operation. Whether the system will recover from a particular finite-time disturbance to a desired stable equilibrium point depends on uncertain and time-varying system parameter values. Therefore, it is valuable to determine, for specific fixed disturbances, the margins for safe operation: the smallest change in parameter values that would cause the system to become vulnerable to the disturbance. The natural setting for this problem is a parameter-dependent vector field with a family of stable equilibria and a parameter-dependent initial condition representing the disturbance. The system recovers for a particular parameter value if its initial condition lies within the region of attraction of the desired stable equilibrium point. Prior work has developed algorithms for numerically computing the margins for safe operation. However, the theoretical guarantees provided for these methods require a very restrictive assumption: that the nonwandering set in the region of attraction boundary is stable under perturbations to the vector field. This assumption is generally intractable to verify, so feasibility of the above algorithms cannot be determined in advance, and even when these algorithms do converge their convergence to the correct values cannot be guaranteed. Thus, this assumption limits the effective application of these algorithms in practice. This work relaxes this restrictive assumption while still obtaining similar results under weaker assumptions, thereby guaranteeing effectiveness of these algorithms. For the setting under consideration, it is shown for vector fields on compact Riemannian manifolds that the restrictive assumption follows immediately and does not need to be independently verified. A motivating example shows that this is not the case for vector fields on Euclidean space, but in this setting it is shown that the restrictive assumption can still be relaxed provided there exist a neighborhood of infinity with suitable properties and some additional generic assumptions. These results are then used to provide theoretical guarantees for the numerical algorithms discussed above under far weaker assumptions.
{"title":"Stability of the Nonwandering Set in the Region of Attraction Boundary under Perturbations with Application to Vulnerability Assessment","authors":"Michael W. Fisher, Ian A. Hiskens","doi":"10.1137/23m155582x","DOIUrl":"https://doi.org/10.1137/23m155582x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3390-3430, December 2023. <br/> Abstract. For many engineered systems it is important to assess vulnerability to potential disturbances in order to ensure reliable operation. Whether the system will recover from a particular finite-time disturbance to a desired stable equilibrium point depends on uncertain and time-varying system parameter values. Therefore, it is valuable to determine, for specific fixed disturbances, the margins for safe operation: the smallest change in parameter values that would cause the system to become vulnerable to the disturbance. The natural setting for this problem is a parameter-dependent vector field with a family of stable equilibria and a parameter-dependent initial condition representing the disturbance. The system recovers for a particular parameter value if its initial condition lies within the region of attraction of the desired stable equilibrium point. Prior work has developed algorithms for numerically computing the margins for safe operation. However, the theoretical guarantees provided for these methods require a very restrictive assumption: that the nonwandering set in the region of attraction boundary is stable under perturbations to the vector field. This assumption is generally intractable to verify, so feasibility of the above algorithms cannot be determined in advance, and even when these algorithms do converge their convergence to the correct values cannot be guaranteed. Thus, this assumption limits the effective application of these algorithms in practice. This work relaxes this restrictive assumption while still obtaining similar results under weaker assumptions, thereby guaranteeing effectiveness of these algorithms. For the setting under consideration, it is shown for vector fields on compact Riemannian manifolds that the restrictive assumption follows immediately and does not need to be independently verified. A motivating example shows that this is not the case for vector fields on Euclidean space, but in this setting it is shown that the restrictive assumption can still be relaxed provided there exist a neighborhood of infinity with suitable properties and some additional generic assumptions. These results are then used to provide theoretical guarantees for the numerical algorithms discussed above under far weaker assumptions.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138552725","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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