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Regular and Chaotic Dynamics最新文献

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On the Uniqueness of Convex Central Configurations in the Planar (4)-Body Problem 平面体问题凸中心构型的唯一性
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-07-31 DOI: 10.1134/S1560354723520076
Shanzhong Sun, Zhifu Xie, Peng You
{"title":"On the Uniqueness of Convex Central Configurations in the Planar (4)-Body Problem","authors":"Shanzhong Sun,&nbsp;Zhifu Xie,&nbsp;Peng You","doi":"10.1134/S1560354723520076","DOIUrl":"10.1134/S1560354723520076","url":null,"abstract":"<div><p>In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture that in the planar four-body problem there exists a unique convex central configuration for any four fixed positive masses in a given order belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem (IFT). Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its neighborhood.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"512 - 532"},"PeriodicalIF":1.4,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50528523","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
Aubry Set on Infinite Cyclic Coverings 无限循环覆盖上的Aubry集
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-07-31 DOI: 10.1134/S1560354723520015
Albert Fathi, Pierre Pageault
{"title":"Aubry Set on Infinite Cyclic Coverings","authors":"Albert Fathi,&nbsp;Pierre Pageault","doi":"10.1134/S1560354723520015","DOIUrl":"10.1134/S1560354723520015","url":null,"abstract":"<div><p>In this paper, we study the projected Aubry set of a lift of a Tonelli\u0000Lagrangian <span>(L)</span> defined on the tangent bundle of a compact manifold <span>(M)</span> to an infinite cyclic covering of <span>(M)</span>. Most of weak KAM and Aubry – Mather theory can be done in this setting. We give a necessary and sufficient condition for the emptiness of the projected Aubry set of the lifted Lagrangian involving both Mather minimizing measures and Mather classes of <span>(L)</span>. Finally, we give Mañè examples on the two-dimensional torus showing that our results do not necessarily hold when the cover is not infinite cyclic.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"425 - 446"},"PeriodicalIF":1.4,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50528657","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
From (2N) to Infinitely Many Escape Orbits 从（2N）到无穷多逃逸轨道
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-07-31 DOI: 10.1134/S1560354723520039
Josep Fontana-McNally, Eva Miranda, Cédric Oms, Daniel Peralta-Salas
{"title":"From (2N) to Infinitely Many Escape Orbits","authors":"Josep Fontana-McNally,&nbsp;Eva Miranda,&nbsp;Cédric Oms,&nbsp;Daniel Peralta-Salas","doi":"10.1134/S1560354723520039","DOIUrl":"10.1134/S1560354723520039","url":null,"abstract":"<div><p>In this short note, we prove that singular Reeb vector fields associated with generic <span>(b)</span>-contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) <span>(2N)</span> or an infinite number of escape orbits, where <span>(N)</span> denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of <span>(b)</span>-Beltrami vector fields that are not <span>(b)</span>-Reeb. The proof is based on a more detailed analysis of the main result in [19].</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"498 - 511"},"PeriodicalIF":1.4,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50528522","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
Total Collision with Slow Convergence to a Degenerate Central Configuration 退化中心构型的慢收敛全碰撞
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-07-31 DOI: 10.1134/S1560354723040020
Richard Moeckel
{"title":"Total Collision with Slow Convergence to a Degenerate Central Configuration","authors":"Richard Moeckel","doi":"10.1134/S1560354723040020","DOIUrl":"10.1134/S1560354723040020","url":null,"abstract":"<div><p>For total collision solutions of the <span>(n)</span>-body problem, Chazy showed that the overall size of the configuration converges to zero with asymptotic rate proportional to <span>(|T-t|^{frac{2}{3}})</span> where <span>(T)</span> is the\u0000collision time. He also showed that the shape of the configuration converges to the set of\u0000central configurations. If the limiting central configuration is nondegenerate, the rate of convergence of the shape is of order <span>(O(|T-t|^{p}))</span> for some <span>(p&gt;0)</span>. Here we show by example that in the planar four-body\u0000problem there exist total collision solutions whose shape converges to a degenerate central configuration at a rate which is slower that any power of <span>(|T-t|)</span>.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"533 - 542"},"PeriodicalIF":1.4,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1134/S1560354723040020.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50528524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Brake Orbits Fill the N-Body Hill Region 制动器轨道填充N型车身坡道区域
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-07-31 DOI: 10.1134/S1560354723520027
Richard Montgomery
{"title":"Brake Orbits Fill the N-Body Hill Region","authors":"Richard Montgomery","doi":"10.1134/S1560354723520027","DOIUrl":"10.1134/S1560354723520027","url":null,"abstract":"<div><p>A brake orbit for the N-body problem is a solution for which, at some instant,\u0000all velocities of all bodies are zero. We reprove two “lost theorems” regarding brake orbits and use them to establish some surprising properties of the completion of the\u0000Jacobi – Maupertuis metric for the N-body problem at negative energies.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"374 - 394"},"PeriodicalIF":1.4,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50528651","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
Integrable Systems on a Sphere, an Ellipsoid and a Hyperboloid 球面、椭球面和超抛物面上的积分系统
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-07-31 DOI: 10.1134/S1560354723520088
Andrey V. Tsiganov
{"title":"Integrable Systems on a Sphere, an Ellipsoid and a Hyperboloid","authors":"Andrey V. Tsiganov","doi":"10.1134/S1560354723520088","DOIUrl":"10.1134/S1560354723520088","url":null,"abstract":"<div><p>Affine transformations in Euclidean space generate a correspondence between integrable systems\u0000on cotangent bundles to a sphere, ellipsoid and hyperboloid embedded in <span>(R^{n})</span>. Using this\u0000correspondence and the suitable coupling constant transformations, we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 6","pages":"805 - 821"},"PeriodicalIF":0.8,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84351670","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
Complex Arnol’d – Liouville Maps 复杂的阿诺尔-刘维尔地图
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-07-31 DOI: 10.1134/S1560354723520064
Luca Biasco, Luigi Chierchia
{"title":"Complex Arnol’d – Liouville Maps","authors":"Luca Biasco,&nbsp;Luigi Chierchia","doi":"10.1134/S1560354723520064","DOIUrl":"10.1134/S1560354723520064","url":null,"abstract":"<div><p>We discuss the holomorphic properties of the complex continuation of the classical Arnol’d – Liouville action-angle variables for real analytic 1 degree-of-freedom Hamiltonian systems depending\u0000on external parameters in suitable Generic Standard Form, with particular regard to the behaviour near separatrices.\u0000In particular, we show that near separatrices the actions, regarded as functions of the energy, have a special universal representation in terms of affine functions of the logarithm with coefficients\u0000analytic functions.\u0000Then, we study the analyticity radii of the action-angle variables in arbitrary neighborhoods of separatrices and describe their behaviour in terms of a (suitably rescaled) distance from separatrices.\u0000Finally, we investigate\u0000the convexity of the energy functions (defined as the inverse of the action functions) near separatrices, and prove that, in particular cases (in the outer regions outside the main separatrix, and in the case the potential is close to a cosine), the convexity is strictly defined, while in general it can be shown that inside separatrices there are inflection points.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"395 - 424"},"PeriodicalIF":1.4,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50528649","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
Emergence of Strange Attractors from Singularities 奇异性中奇异吸引子的产生
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-07-31 DOI: 10.1134/S1560354723520040
José Angel Rodríguez
{"title":"Emergence of Strange Attractors from Singularities","authors":"José Angel Rodríguez","doi":"10.1134/S1560354723520040","DOIUrl":"10.1134/S1560354723520040","url":null,"abstract":"<div><p>This paper is a summary of results that prove the abundance of\u0000one-dimensional strange attractors near a Shil’nikov configuration, as well\u0000as the presence of these configurations in generic unfoldings of\u0000singularities in <span>(mathbb{R}^{3})</span> of minimal codimension.\u0000Finding these singularities in families of vector fields is analytically possible and thus provides a tractable criterion for the existence of chaotic dynamics.\u0000Alternative scenarios for the possible abundance of two-dimensional attractors in higher\u0000dimension are also presented. The role of Shil’nikov configuration is now played by a certain type of generalised tangency which should occur for families of vector fields <span>(X_{mu})</span>\u0000unfolding generically some low codimension singularity in <span>(mathbb{R}^{n})</span>\u0000with <span>(ngeqslant 4)</span>.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"468 - 497"},"PeriodicalIF":1.4,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50528521","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
Attractive Invariant Circles à la Chenciner La Chenciner的吸引人的不变圆
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-07-31 DOI: 10.1134/S1560354723520052
Jessica Elisa Massetti
{"title":"Attractive Invariant Circles à la Chenciner","authors":"Jessica Elisa Massetti","doi":"10.1134/S1560354723520052","DOIUrl":"10.1134/S1560354723520052","url":null,"abstract":"<div><p>In studying general perturbations of a dissipative twist map depending on two parameters, a frequency <span>(nu)</span> and a dissipation <span>(eta)</span>, the existence of a Cantor set <span>(mathcal{C})</span> of curves in the <span>((nu,eta))</span> plane such that the corresponding equation possesses a Diophantine quasi-periodic invariant circle can be deduced, up to small values of the dissipation, as a direct consequence of a normal form theorem in the spirit of Rüssmann and the “elimination of parameters” technique. These circles are normally hyperbolic as soon as <span>(etanot=0)</span>, which implies that the equation still possesses a circle of this kind for values of the parameters belonging to a neighborhood <span>(mathcal{V})</span> of this set of curves. Obviously, the dynamics on such invariant circles is no more controlled and may be generic, but the normal dynamics is controlled in the sense of their basins of attraction.</p><p>As expected, by the classical graph-transform method we are able to determine a first rough region where the normal hyperbolicity prevails and a circle persists, for a strong enough dissipation <span>(etasim O(sqrt{varepsilon}),)</span> <span>(varepsilon)</span> being the size of the perturbation. Then, through normal-form techniques, we shall enlarge such regions and determine such a (conic) neighborhood <span>(mathcal{V})</span>, up to values of dissipation of the same order as the perturbation, by using the fact that the proximity of the set <span>(mathcal{C})</span>\u0000allows, thanks to Rüssmann’s translated curve theorem, an introduction of local coordinates of the type (dissipation, translation) similar to the ones introduced by Chenciner in [7].</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"447 - 467"},"PeriodicalIF":1.4,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50528658","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 Partially Hyperbolic Diffeomorphisms and Regular Denjoy Type Homeomorphisms 部分双曲微分同态和正则Denjoy型同态
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-06-02 DOI: 10.1134/S1560354723030036
Vyacheslav Z. Grines, Dmitrii I. Mints
{"title":"On Partially Hyperbolic Diffeomorphisms and Regular Denjoy Type Homeomorphisms","authors":"Vyacheslav Z. Grines,&nbsp;Dmitrii I. Mints","doi":"10.1134/S1560354723030036","DOIUrl":"10.1134/S1560354723030036","url":null,"abstract":"<div><p>In P. D. McSwiggen’s article, it was proposed Derived from Anosov type construction which leads to a partially hyperbolic diffeomorphism of the 3-torus. The nonwandering set of this diffeomorphism contains a two-dimensional attractor which consists of one-dimensional unstable manifolds of its points. The constructed\u0000diffeomorphism admits an invariant one-dimensional orientable foliation such that it contains\u0000unstable manifolds of points of the attractor as its leaves. Moreover, this foliation has a\u0000global cross section (2-torus) and defines on it a Poincaré map which is a regular Denjoy\u0000type homeomorphism. Such homeomorphisms are the most natural generalization of Denjoy\u0000homeomorphisms of the circle and play an important role in the description of the dynamics\u0000of aforementioned partially hyperbolic diffeomorphisms. In particular, the topological\u0000conjugacy of corresponding Poincaré maps provides necessary conditions for the topological\u0000conjugacy of the restrictions of such partially hyperbolic diffeomorphisms to\u0000their two-dimensional attractors. The nonwandering set of each regular Denjoy type homeomorphism\u0000is a Sierpiński set and each such homeomorphism is, by definition, semiconjugate to the\u0000minimal translation of the 2-torus. We introduce a complete invariant of topological conjugacy\u0000for regular Denjoy type homeomorphisms that is characterized by the minimal translation,\u0000which is semiconjugation of the given regular Denjoy type homeomorphism, with a distinguished,\u0000no more than countable set of orbits.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 3","pages":"295 - 308"},"PeriodicalIF":1.4,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4090578","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
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