{"title":"Fast Adjoint Algorithm for Linear Responses of Hyperbolic Chaos","authors":"Angxiu Ni","doi":"10.1137/22m1522383","DOIUrl":"https://doi.org/10.1137/22m1522383","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"248 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135970091","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":"Equilibria Analysis of a Networked Bivirus Epidemic Model Using Poincaré–Hopf and Manifold Theory","authors":"Brian D. O. Anderson, Mengbin Ye","doi":"10.1137/22m1529981","DOIUrl":"https://doi.org/10.1137/22m1529981","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135969108","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}
The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace of and its dual space . The three topological spaces are called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron–Frobenius operators of the one-sided and two-sided shifts of finite type (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ratio on modulo 1 is also considered. A new construction of the Gelfand triplet for the generalized spectrum of symbolic dynamical systems is proposed by means of an algebraic procedure. The asymptotic formula of the iteration of Perron–Frobenius operators is also given. The iteration converges to the mixing state whose rate of convergence is determined by the generalized spectrum.
{"title":"Generalized Eigenvalues of the Perron–Frobenius Operators of Symbolic Dynamical Systems","authors":"Hayato Chiba, Masahiro Ikeda, Isao Ishikawa","doi":"10.1137/22m1476204","DOIUrl":"https://doi.org/10.1137/22m1476204","url":null,"abstract":"The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace of and its dual space . The three topological spaces are called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron–Frobenius operators of the one-sided and two-sided shifts of finite type (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ratio on modulo 1 is also considered. A new construction of the Gelfand triplet for the generalized spectrum of symbolic dynamical systems is proposed by means of an algebraic procedure. The asymptotic formula of the iteration of Perron–Frobenius operators is also given. The iteration converges to the mixing state whose rate of convergence is determined by the generalized spectrum.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135969467","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}
Mauricio Ascencio, Esther Barrabés, Josep M. Cors, Claudio Vidal
We study the dynamics of an infinitesimal mass under the gravitational attraction of primaries arranged in a planar ring configuration plus the influence of the central mass with a Manev potential , , where is a parameter related to the oblaticity or radiation source (according to the sign of the parameter ). Specifically, we investigate the relative equilibria of the infinitesimal mass and their linear stability as functions of the parameter and the mass parameter , the ratio of mass of the central body to the mass of one of the remaining bodies. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.
{"title":"Stability of Equilibrium Points in the Spatially Restricted ({boldsymbol{N+1}}) -Body Problem with Manev Potential","authors":"Mauricio Ascencio, Esther Barrabés, Josep M. Cors, Claudio Vidal","doi":"10.1137/23m1551912","DOIUrl":"https://doi.org/10.1137/23m1551912","url":null,"abstract":"We study the dynamics of an infinitesimal mass under the gravitational attraction of primaries arranged in a planar ring configuration plus the influence of the central mass with a Manev potential , , where is a parameter related to the oblaticity or radiation source (according to the sign of the parameter ). Specifically, we investigate the relative equilibria of the infinitesimal mass and their linear stability as functions of the parameter and the mass parameter , the ratio of mass of the central body to the mass of one of the remaining bodies. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136209243","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":"Optimal Control of the Controlled Lotka–Volterra Equations with Applications. The Permanent Case","authors":"Bernard Bonnard, Jérémy Rouot","doi":"10.1137/22m151978x","DOIUrl":"https://doi.org/10.1137/22m151978x","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136058087","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}
Pedro Parra-Rivas, Alan R. Champneys, Fahad Al Saadi, Damia Gomila, Edgar Knobloch
A wide variety of stationary or moving spatially localized structures is present in evolution problems on unbounded domains, governed by higher-than-second-order reversible spatial interactions. This work provides a generic unfolding in one spatial dimension of a certain codimension-three singularity that explains the organization of bifurcation diagrams of such localized states in a variety of contexts, ranging from nonlinear optics to fluid mechanics, mathematical biology, and beyond. The singularity occurs when a cusp bifurcation associated with the onset of bistability between homogeneous steady states encounters a pattern-forming, or Turing, bifurcation. The latter corresponds to a Hamiltonian-Hopf point of the corresponding spatial dynamics problem. Such codimension-three points are sometimes called Lifshitz points in the physics literature. In the simplest case where the spatial system conserves a first integral, the system is described by a canonical fourth-order scalar system. The problem contains three small parameters: two that unfold the cusp bifurcation and one that unfolds the Turing bifurcation. Several cases are revealed, depending on open conditions on the signs of the lowest-order nonlinear terms. Taking the case in which the Turing bifurcation is subcritical, various parameter regimes are considered and the bifurcation diagrams of localized structures are elucidated. A rich bifurcation structure is revealed which involves transitions between regions of localized periodic patterns generated by standard homoclinic snaking, and regions of stationary domains of one homogeneous solution embedded in the other organized in a collapsed snaking structure. The theory is shown to unify previous numerical results obtained in models arising in nonlinear optics, fluid mechanics, and excitable media more generally.
{"title":"Organization of Spatially Localized Structures near a Codimension-Three Cusp-Turing Bifurcation","authors":"Pedro Parra-Rivas, Alan R. Champneys, Fahad Al Saadi, Damia Gomila, Edgar Knobloch","doi":"10.1137/22m1514234","DOIUrl":"https://doi.org/10.1137/22m1514234","url":null,"abstract":"A wide variety of stationary or moving spatially localized structures is present in evolution problems on unbounded domains, governed by higher-than-second-order reversible spatial interactions. This work provides a generic unfolding in one spatial dimension of a certain codimension-three singularity that explains the organization of bifurcation diagrams of such localized states in a variety of contexts, ranging from nonlinear optics to fluid mechanics, mathematical biology, and beyond. The singularity occurs when a cusp bifurcation associated with the onset of bistability between homogeneous steady states encounters a pattern-forming, or Turing, bifurcation. The latter corresponds to a Hamiltonian-Hopf point of the corresponding spatial dynamics problem. Such codimension-three points are sometimes called Lifshitz points in the physics literature. In the simplest case where the spatial system conserves a first integral, the system is described by a canonical fourth-order scalar system. The problem contains three small parameters: two that unfold the cusp bifurcation and one that unfolds the Turing bifurcation. Several cases are revealed, depending on open conditions on the signs of the lowest-order nonlinear terms. Taking the case in which the Turing bifurcation is subcritical, various parameter regimes are considered and the bifurcation diagrams of localized structures are elucidated. A rich bifurcation structure is revealed which involves transitions between regions of localized periodic patterns generated by standard homoclinic snaking, and regions of stationary domains of one homogeneous solution embedded in the other organized in a collapsed snaking structure. The theory is shown to unify previous numerical results obtained in models arising in nonlinear optics, fluid mechanics, and excitable media more generally.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136295577","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":"Critical Transitions in d-Concave Nonautonomous Scalar Ordinary Differential Equations Appearing in Population Dynamics","authors":"Jesús Dueñas, Carmen Núñez, Rafael Obaya","doi":"10.1137/22m1542830","DOIUrl":"https://doi.org/10.1137/22m1542830","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136295785","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":"Non-Markovian Models of Opinion Dynamics on Temporal Networks","authors":"Weiqi Chu, Mason A. Porter","doi":"10.1137/22m151858x","DOIUrl":"https://doi.org/10.1137/22m151858x","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135150588","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":"Abundance of Infinite Switching","authors":"Alexandre A. P. Rodrigues, Luísa Castro","doi":"10.1137/22m151371x","DOIUrl":"https://doi.org/10.1137/22m151371x","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"363 1","pages":"2570-2600"},"PeriodicalIF":2.1,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64316545","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":"Stability Analysis of the Immune System Induced by Chemotaxis","authors":"Pan Zheng, Wenhai Shan, Guangyuan Liao","doi":"10.1137/22m1498310","DOIUrl":"https://doi.org/10.1137/22m1498310","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"22 1","pages":"2527-2569"},"PeriodicalIF":2.1,"publicationDate":"2023-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64316256","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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