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Regular and Chaotic Dynamics最新文献

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Construction of the Morse – Bott Energy Function for Regular Topological Flows 正则拓扑流的Morse - Bott能量函数的构造
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2021-08-09 DOI: 10.1134/S1560354721040031
Olga V. Pochinka, Svetlana Kh. Zinina
{"title":"Construction of the Morse – Bott Energy Function for Regular Topological Flows","authors":"Olga V. Pochinka,&nbsp;Svetlana Kh. Zinina","doi":"10.1134/S1560354721040031","DOIUrl":"10.1134/S1560354721040031","url":null,"abstract":"<div><p>In this paper, we consider regular topological flows on closed <span>(n)</span>-manifolds. Such flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale flows, which are closely related to the topology of the supporting manifold. This connection is provided by the existence of the Morse – Bott energy function for the Morse – Smale flows. It is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds, on which dynamical systems can be considered only in a continuous category. The existence of continuous analogs of regular flows on any topological manifolds is an open question, as is the existence of energy functions for such flows. In this paper, we study the dynamics of regular topological flows, investigate the topology of the embedding and the asymptotic behavior of invariant manifolds of fixed points and periodic orbits. The main result is the construction of the Morse – Bott energy function for such flows, which ensures their close connection with the topology of the ambient manifold.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 4","pages":"350 - 369"},"PeriodicalIF":1.4,"publicationDate":"2021-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4680131","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}
引用次数: 3
Sections of Hamiltonian Systems 哈密顿系统的部分
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2021-08-09 DOI: 10.1134/S156035472104002X
Konstantinos Kourliouros
{"title":"Sections of Hamiltonian Systems","authors":"Konstantinos Kourliouros","doi":"10.1134/S156035472104002X","DOIUrl":"10.1134/S156035472104002X","url":null,"abstract":"<div><p>A section of a Hamiltonian system is a hypersurface in the phase space of the system,\u0000usually representing a set of one-sided constraints (e. g., a boundary, an obstacle or a set\u0000of admissible states). In this paper we give local classification results for all typical singularities of sections of regular (non-singular) Hamiltonian systems, a problem equivalent to the classification of typical singularities of Hamiltonian systems with one-sided constraints. In particular, we give a complete list of\u0000exact normal forms with functional invariants, and we show how these are related/obtained by the symplectic classification of mappings with prescribed (Whitney-type) singularities, naturally defined on the reduced phase space of the Hamiltonian system.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 4","pages":"331 - 349"},"PeriodicalIF":1.4,"publicationDate":"2021-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4376039","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}
引用次数: 0
On Singularly Perturbed Linear Cocycles over Irrational Rotations 关于非理性旋转上的奇摄动线性环
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2021-06-03 DOI: 10.1134/S1560354721030011
Alexey V. Ivanov
{"title":"On Singularly Perturbed Linear Cocycles over Irrational Rotations","authors":"Alexey V. Ivanov","doi":"10.1134/S1560354721030011","DOIUrl":"10.1134/S1560354721030011","url":null,"abstract":"<div><p>We study a linear cocycle over the irrational rotation <span>(sigma_{omega}(x)=x+omega)</span> of the circle <span>(mathbb{T}^{1})</span>. It is supposed that the cocycle is generated by a <span>(C^{2})</span>-map\u0000<span>(A_{varepsilon}:mathbb{T}^{1}to SL(2,mathbb{R}))</span> which depends on a small parameter <span>(varepsilonll 1)</span> and has the form of the Poincaré map corresponding to a singularly perturbed Hill equation with quasi-periodic potential. Under the assumption that the norm of the matrix <span>(A_{varepsilon}(x))</span> is of order <span>(exp(pmlambda(x)/varepsilon))</span>, where <span>(lambda(x))</span> is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter <span>(varepsilon)</span>. We show that in the limit <span>(varepsilonto 0)</span> the cocycle “typically” exhibits ED only if it is exponentially close to a constant cocycle.\u0000Conversely, if the cocycle is not close to a constant one,\u0000it does not possess ED, whereas the Lyapunov exponent is “typically” large.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 3","pages":"205 - 221"},"PeriodicalIF":1.4,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4128116","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}
引用次数: 4
Generic Properties of Mañé’s Set of Exact Magnetic Lagrangians Mañé精确磁拉格朗日集的一般性质
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2021-06-03 DOI: 10.1134/S1560354721030060
Alexandre Rocha
{"title":"Generic Properties of Mañé’s Set of Exact Magnetic Lagrangians","authors":"Alexandre Rocha","doi":"10.1134/S1560354721030060","DOIUrl":"10.1134/S1560354721030060","url":null,"abstract":"<div><p>Let <span>(M)</span> be a closed manifold and <span>(L)</span> an exact magnetic Lagrangian. In this\u0000paper we prove that there exists a residual set <span>(mathcal{G})</span> of <span>(H^{1}left(M;mathbb{R}right))</span> such that the property\u0000</p><div><div><span>\u0000$${widetilde{mathcal{M}}}left(cright)={widetilde{mathcal{A}}}left(cright)={widetilde{mathcal{N}}}left(cright),forall cinmathcal{G},$$\u0000</span></div></div><p>\u0000with <span>({widetilde{mathcal{M}}}left(cright))</span> supporting a uniquely\u0000ergodic measure, is generic in the family of exact magnetic Lagrangians. We also prove\u0000that, for a fixed cohomology class <span>(c)</span>, there exists a\u0000residual set of exact magnetic Lagrangians such that, when this\u0000unique\u0000measure is supported on a periodic orbit, this orbit is hyperbolic and its\u0000stable and unstable manifolds intersect transversally. This result is a\u0000version of an analogous theorem, for Tonelli Lagrangians, proven in [6].</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 3","pages":"293 - 304"},"PeriodicalIF":1.4,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4129773","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}
引用次数: 0
On the Isolation/Nonisolation of a Cosymmetric Equilibrium and Bifurcations in its Neighborhood 一类协对称平衡及其邻域分岔的孤立/非孤立性
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2021-06-03 DOI: 10.1134/S1560354721030047
Leonid G. Kurakin, Aik V. Kurdoglyan
{"title":"On the Isolation/Nonisolation of a Cosymmetric Equilibrium and Bifurcations in its Neighborhood","authors":"Leonid G. Kurakin,&nbsp;Aik V. Kurdoglyan","doi":"10.1134/S1560354721030047","DOIUrl":"10.1134/S1560354721030047","url":null,"abstract":"<div><p>A dynamical system with a cosymmetry is considered.\u0000V. I. Yudovich showed that a noncosymmetric\u0000equilibrium of such a system under the conditions of\u0000the general position is a member of a one-parameter\u0000family.\u0000In this paper, it is assumed that the equilibrium is\u0000cosymmetric, and the linearization matrix of the\u0000cosymmetry is nondegenerate.\u0000It is shown that, in the case of an odd-dimensional\u0000dynamical system, the equilibrium is also nonisolated\u0000and belongs to a one-parameter family of equilibria.\u0000In the even-dimensional case, the cosymmetric equilibrium is,\u0000generally speaking, isolated.\u0000The Lyapunov – Schmidt method is used to study bifurcations\u0000in the neighborhood of the cosymmetric equilibrium when\u0000the linearization matrix has a double kernel.\u0000The dynamical system and its cosymmetry depend on a real\u0000parameter.\u0000We describe scenarios of branching for families of\u0000noncosymmetric equilibria.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 3","pages":"258 - 270"},"PeriodicalIF":1.4,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4129123","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}
引用次数: 0
Recruitment Effects on the Evolution of Epidemics in a Simple SIR Model 在简单SIR模型中招募效应对流行病演化的影响
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2021-06-03 DOI: 10.1134/S1560354721030072
Gilberto Nakamura, Basil Grammaticos, Mathilde Badoual
{"title":"Recruitment Effects on the Evolution of Epidemics in a Simple SIR Model","authors":"Gilberto Nakamura,&nbsp;Basil Grammaticos,&nbsp;Mathilde Badoual","doi":"10.1134/S1560354721030072","DOIUrl":"10.1134/S1560354721030072","url":null,"abstract":"<div><p>We analyse the patterns of the current epidemic evolution in various countries with the help of a simple SIR model. We consider two main effects: climate induced seasonality and recruitment. The latter is introduced as a way to palliate for the absence of a spatial component in the SIR model. In our approach we mimic the spatial evolution of the epidemic through a gradual introduction of susceptible individuals.\u0000</p><p>\u0000We apply our model to the case of France and Australia and explain the appearance of two temporally well-separated epidemic waves. We examine also Brazil and the USA, which present patterns very different from those of the European countries. We show that with our model it is possible to reproduce the observed patterns in these two countries thanks to simple recruitment assumptions. Finally, in order to show the power of the recruitment approach, we simulate the case of the 1918 influenza epidemic reproducing successfully the, by now famous, three epidemic peaks.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 3","pages":"305 - 319"},"PeriodicalIF":1.4,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4130062","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}
引用次数: 2
A Restricted Four-body Problem for the Figure-eight Choreography 八字舞蹈的限制性四体问题
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2021-06-03 DOI: 10.1134/S1560354721030023
Ricardo Lara, Abimael Bengochea
{"title":"A Restricted Four-body Problem for the Figure-eight Choreography","authors":"Ricardo Lara,&nbsp;Abimael Bengochea","doi":"10.1134/S1560354721030023","DOIUrl":"10.1134/S1560354721030023","url":null,"abstract":"<div><p>In this work we introduce a planar restricted four-body problem where a massless particle moves under the gravitational influence due to three bodies following the figure-eight choreography, and explore some symmetric periodic orbits of this system which turns out to be nonautonomous. We use reversing symmetries to study both theoretically and numerically a certain type of symmetric periodic orbits of this system.\u0000The symmetric periodic orbits (initial conditions) were computed by solving some\u0000boundary-value problems.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 3","pages":"222 - 235"},"PeriodicalIF":1.4,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4129341","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}
引用次数: 0
Lax Pairs and Rational Solutions of Similarity Reductions for Kupershmidt and Sawada – Kotera Hierarchies Kupershmidt和Sawada - Kotera层次相似性约简的Lax对和有理解
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2021-06-03 DOI: 10.1134/S1560354721030059
Nikolay A. Kudryashov
{"title":"Lax Pairs and Rational Solutions of Similarity Reductions for Kupershmidt and Sawada – Kotera Hierarchies","authors":"Nikolay A. Kudryashov","doi":"10.1134/S1560354721030059","DOIUrl":"10.1134/S1560354721030059","url":null,"abstract":"<div><p>Self-similar reductions for equations of the Kupershmidt and Sawada – Kotera hierarchies are\u0000considered. Algorithms for constructing a Lax pair\u0000for equations of these hierarchies are presented. Lax pairs for ordinary differential\u0000equations of the fifth, seventh and eleventh orders\u0000corresponding to the Kupershmidt and the Sawada – Kotera hierarchies are given.\u0000The Lax pairs allow us to solve these equations by means of the inverse\u0000monodromy transform method. The application of the Painlevé test to the seventh order of the similarity reduction for the Kupershmidt hierarchy is\u0000demonstrated. It is shown that special solutions of the similarity reductions for the Kupershnmidt and Sawada – Kotera hierarchies are determined via\u0000the transcendents of the <span>(K_{1})</span> and <span>(K_{2})</span> hierarchies. Rational solutions of the similarity reductions of the modified Kupershmidt and Sawada – Kotera\u0000hierarchies are given. Special polynomials associated with the self-similar reductions of\u0000the Kupershmidt and Sawada – Kotera hierarchies are presented.\u0000Rational solutions of some hierarchies are calculated by means of the Miura transformations and taking into account special polynomials.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 3","pages":"271 - 292"},"PeriodicalIF":1.4,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4129772","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}
引用次数: 2
Compactness and Index of Ordinary Central Configurations for the Curved (N)-Body Problem 弯曲(N) -体问题普通中心构型的紧性和指数
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2021-06-03 DOI: 10.1134/S1560354721030035
Shuqiang Zhu
{"title":"Compactness and Index of Ordinary Central Configurations for the Curved (N)-Body Problem","authors":"Shuqiang Zhu","doi":"10.1134/S1560354721030035","DOIUrl":"10.1134/S1560354721030035","url":null,"abstract":"<div><p>For the curved <span>(n)</span>-body problem, we show that the set of ordinary central configurations is away from singular configurations in <span>(mathbb{H}^{3})</span> with positive momentum of inertia, and away from a subset of singular\u0000configurations in <span>(mathbb{S}^{3})</span>. We also show that\u0000each of the <span>(n!/2)</span> geodesic ordinary central configurations for <span>(n)</span> masses has Morse index <span>(n-2)</span>.\u0000Then we get a direct corollary that there are at least <span>(frac{(3n-4)(n-1)!}{2})</span> ordinary central\u0000configurations for given <span>(n)</span> masses if all ordinary central configurations of these masses are nondegenerate.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 3","pages":"236 - 257"},"PeriodicalIF":1.4,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4130061","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}
引用次数: 3
Classical and Quantum Dynamical Manifestations ofIndex-(2) Saddles: Concerted Versus Sequential Reaction Mechanisms findex - (2)鞍的经典和量子动力学表现:协调与顺序反应机制
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2021-04-05 DOI: 10.1134/S1560354721020052
Priyanka Pandey, Shibabrat Naik, Srihari Keshavamurthy
{"title":"Classical and Quantum Dynamical Manifestations of\u0000Index-(2) Saddles: Concerted Versus Sequential Reaction Mechanisms","authors":"Priyanka Pandey,&nbsp;Shibabrat Naik,&nbsp;Srihari Keshavamurthy","doi":"10.1134/S1560354721020052","DOIUrl":"https://doi.org/10.1134/S1560354721020052","url":null,"abstract":"<p>The presence of higher-index saddles on a multidimensional potential energy surface is usually assumed to be of little significance in chemical reaction dynamics. Such a viewpoint requires careful reconsideration, thanks to elegant experiments and novel theoretical approaches that have come about in recent years. In this work, we perform a detailed classical and quantum dynamical study of a model two-degree-of-freedom Hamiltonian, which captures the essence of the debate regarding the dominance of a concerted or a stepwise reaction mechanism. We show that the ultrafast shift of the mechanism from a concerted to a stepwise one is essentially a classical dynamical effect. In addition, due to the classical phase space being a mixture of regular and chaotic dynamics, it is possible to have a rich variety of dynamical behavior, including a Murrell – Laidler type mechanism, even at energies sufficiently above that of the index-<span>(2)</span> saddle. We rationalize the dynamical results using an explicit construction of the classical invariant manifolds in the phase space.</p>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 2","pages":"165 - 182"},"PeriodicalIF":1.4,"publicationDate":"2021-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4529566","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}
引用次数: 1
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