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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
Formal Stability, Stability for Most Initial Conditions and Diffusion in Analytic Systems of Differential Equations 形式稳定性，大多数初始条件的稳定性和微分方程解析系统的扩散
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-06-02 DOI: 10.1134/S1560354723030012
Valery V. Kozlov
{"title":"Formal Stability, Stability for Most Initial Conditions and Diffusion in Analytic Systems of Differential Equations","authors":"Valery V. Kozlov","doi":"10.1134/S1560354723030012","DOIUrl":"10.1134/S1560354723030012","url":null,"abstract":"<div><p>An example of an analytic system of differential equations in <span>(mathbb{R}^{6})</span> with an equilibrium\u0000formally stable and stable for most initial conditions is presented. By means of a divergent formal transformation this system is reduced to a Hamiltonian system with three degrees of freedom. Almost all its phase space is foliated by three-dimensional invariant tori carrying quasi-periodic trajectories.\u0000These tori do not fill all phase space. Though the “gap” between these tori has zero measure, this set is everywhere dense in <span>(mathbb{R}^{6})</span> and unbounded phase trajectories are dense in this gap. In particular, the formally stable equilibrium is Lyapunov unstable. This behavior of phase trajectories is quite consistent with the diffusion in nearly integrable systems. The proofs are based on the Poincaré – Dulac theorem, the theory of almost periodic functions, and on some facts from the theory of inhomogeneous Diophantine approximations. Some open problems related to the example are presented.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 3","pages":"251 - 264"},"PeriodicalIF":1.4,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4090144","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
A Note on the Weighted Yamabe Flow 关于加权Yamabe流的注释
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-06-02 DOI: 10.1134/S1560354723030048
Theodore Yu. Popelensky
{"title":"A Note on the Weighted Yamabe Flow","authors":"Theodore Yu. Popelensky","doi":"10.1134/S1560354723030048","DOIUrl":"10.1134/S1560354723030048","url":null,"abstract":"<div><p>For two dimensional surfaces (smooth) Ricci and Yamabe flows are equivalent.\u0000In 2003, Chow and Luo developed the theory of combinatorial Ricci flow for circle packing metrics on closed triangulated surfaces.\u0000In 2004, Luo developed a theory of discrete Yamabe flow for closed triangulated surfaces.\u0000He investigated the formation of singularities and convergence to a metric of constant curvature.</p><p>In this note we develop the theory of a naïve discrete Ricci flow and its modification — the so-called weighted Ricci flow. We prove that this flow has a rich family of first integrals and is equivalent to a certain modification of Luo’s discrete Yamabe flow.\u0000We investigate the types of singularities of solutions for these flows and discuss convergence to a metric of weighted\u0000constant curvature.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 3","pages":"309 - 320"},"PeriodicalIF":1.4,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4091342","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
V-Shaped Action Functional with Delay 具有延迟的V形作用函数
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-04-10 DOI: 10.1134/S1560354723510020
Urs Frauenfelder
{"title":"V-Shaped Action Functional with Delay","authors":"Urs Frauenfelder","doi":"10.1134/S1560354723510020","DOIUrl":"10.1134/S1560354723510020","url":null,"abstract":"<div><p>In this note we introduce the V-shaped action functional with delay in a symplectization,\u0000which is an intermediate action functional between the Rabinowitz action functional\u0000and the V-shaped action functional. It lives on the same space as the\u0000V-shaped action functional, but its gradient flow equation is a delay equation\u0000as in the case of the Rabinowitz action functional. We show that there is a smooth interpolation\u0000between the V-shaped action functional and the V-shaped action functional with delay\u0000during which the critical points and its actions are fixed. Moreover, we prove that there\u0000is a bijection between gradient flow lines of the V-shaped action functional with delay\u0000and the ones of the Rabinowitz action functional.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"364 - 373"},"PeriodicalIF":1.4,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50467130","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
Twist Maps of the Annulus: An Abstract Point of View 环空的扭曲映射：一个抽象的观点
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-04-10 DOI: 10.1134/S1560354723510019
Patrice Le Calvez
{"title":"Twist Maps of the Annulus: An Abstract Point of View","authors":"Patrice Le Calvez","doi":"10.1134/S1560354723510019","DOIUrl":"10.1134/S1560354723510019","url":null,"abstract":"<div><p>We introduce the notion of abstract angle at a couple of points defined by two radial foliations of the closed annulus. We will use for this purpose the digital line topology on the set <span>({mathbb{Z}})</span> of relative integers, also called the Khalimsky topology. We use this notion to give unified proofs of some classical results on area preserving positive twist maps of the annulus by using the Lifting Theorem and the Intermediate Value Theorem. More precisely, we will interpretate Birkhoff theory about annular invariant open sets in this formalism. Then we give a proof of Mather’s theorem stating the existence of crossing orbits in a Birkhoff region of instability. Finally we will give a proof of Poincaré – Birkhoff theorem in a particular case, that includes the case where the map is a composition of positive twist maps.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"343 - 363"},"PeriodicalIF":1.4,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50467129","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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