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Regular and Chaotic Dynamics最新文献

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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
Numerical and Theoretical Studies on the Rational Standard Map at Moderate-to-Large Values of the Amplitude Parameter 幅值参数中大值的有理标准图的数值与理论研究
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-06-02 DOI: 10.1134/S1560354723030024
Pablo M. Cincotta, Claudia M. Giordano, Carles Simó
{"title":"Numerical and Theoretical Studies on the Rational Standard Map at Moderate-to-Large Values of the Amplitude Parameter","authors":"Pablo M. Cincotta,&nbsp;Claudia M. Giordano,&nbsp;Carles Simó","doi":"10.1134/S1560354723030024","DOIUrl":"10.1134/S1560354723030024","url":null,"abstract":"<div><p>In this work an exhaustive numerical and analytical investigation of the dynamics of a bi-parametric symplectic\u0000map, the so-called rational\u0000standard map, at moderate-to-large values of the\u0000amplitude parameter is addressed. After reviewing the model, a discussion concerning an analytical\u0000determination of the maximum Lyapunov exponent is provided together with thorough numerical experiments.\u0000The theoretical results are obtained in the limit of a nearly uniform distribution of the phase values.\u0000Correlations among phases lead to departures from the expected estimates.\u0000In this direction, a detailed study of the role of stable periodic islands of periods 1, 2 and 4 is included.\u0000Finally, an experimental relationship between the Lyapunov and instability times is shown,\u0000while an analytical one applies when correlations are irrelevant, which is the case, in general,\u0000for large values of the amplitude parameter.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 3","pages":"265 - 294"},"PeriodicalIF":1.4,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4091360","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
Parametric Resonance of a Charged Pendulum with a Suspension Point Oscillating Between Two Vertical Charged Lines 悬点在两条垂直带电线之间振荡的带电摆的参数共振
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-06-02 DOI: 10.1134/S156035472303005X
Adecarlos C. Carvalho, Gerson C. Araujo
{"title":"Parametric Resonance of a Charged Pendulum with a Suspension Point Oscillating Between Two Vertical Charged Lines","authors":"Adecarlos C. Carvalho,&nbsp;Gerson C. Araujo","doi":"10.1134/S156035472303005X","DOIUrl":"10.1134/S156035472303005X","url":null,"abstract":"<div><p>In this study, we analyze a planar mathematical pendulum with a suspension point that oscillates harmonically in the vertical direction. The bob of the pendulum is electrically charged and is located between two wires with a uniform distribution of electric charges, both equidistant from the suspension point. The dynamics of this phenomenon is investigated. The system has three parameters, and we analyze the parametric stability of the equilibrium points, determining surfaces that separate the regions of stability and instability in the parameter space. In the case where the parameter associated with the charges is equal to zero, we obtain boundary curves that separate the regions of stability and instability for the Mathieu equation.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 3","pages":"321 - 331"},"PeriodicalIF":1.4,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4090128","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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