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Regular and Chaotic Dynamics最新文献

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On the Interplay Between Vortices and Harmonic Flows: Hodge Decomposition of Euler’s Equations in 2d 论涡流与谐波流的相互作用：二维欧拉方程的霍奇分解
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-04-08 DOI: 10.1134/S1560354724020011
Clodoaldo Grotta-Ragazzo, Björn Gustafsson, Jair Koiller
{"title":"On the Interplay Between Vortices and Harmonic Flows: Hodge Decomposition of Euler’s Equations in 2d","authors":"Clodoaldo Grotta-Ragazzo,&nbsp;Björn Gustafsson,&nbsp;Jair Koiller","doi":"10.1134/S1560354724020011","DOIUrl":"10.1134/S1560354724020011","url":null,"abstract":"<div><p>Let <span>(Sigma)</span> be a compact manifold without boundary whose first homology is nontrivial. The Hodge decomposition of the incompressible Euler equation in terms of 1-forms yields a coupled PDE-ODE system. The <span>(L^{2})</span>-orthogonal components are a “pure” vorticity flow and a potential flow (harmonic, with the dimension of the homology). In this paper we focus on <span>(N)</span> point vortices on a compact Riemann surface without boundary of genus <span>(g)</span>, with a metric chosen in the conformal class. The phase space has finite dimension <span>(2N+2g)</span>. We compute a surface of section for the motion of a single vortex (<span>(N=1)</span>) on a torus (<span>(g=1)</span>) with a nonflat metric that shows typical features of nonintegrable 2 degrees of freedom Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces (<span>(ggeqslant 2)</span>) having constant curvature <span>(-1)</span>, with discrete symmetries. Fixed points of involutions yield vortex crystals in the Poincaré disk. Finally, we consider multiply connected planar domains. The image method due to Green and Thomson is\u0000viewed in the Schottky double. The Kirchhoff – Routh Hamiltonian\u0000given in C. C. Lin’s celebrated theorem is recovered by\u0000Marsden – Weinstein reduction from <span>(2N+2g)</span> to <span>(2N)</span>.\u0000The relation between the electrostatic Green function and the\u0000hydrodynamic Green function is clarified.\u0000A number of questions are suggested.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 2","pages":"241 - 303"},"PeriodicalIF":0.8,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579141","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 Eisenhart’s Type Theorem for Sub-Riemannian Metrics on Step (2) Distributions with (mathrm{ad})-Surjective Tanaka Symbols 关于带有$$mathrm{ad}$$-Surjective Tanaka符号的阶$$2$分布上子黎曼度量的艾森哈特类型定理
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-04-08 DOI: 10.1134/S1560354724020023
Zaifeng Lin, Igor Zelenko
{"title":"On Eisenhart’s Type Theorem for Sub-Riemannian Metrics on Step (2) Distributions with (mathrm{ad})-Surjective Tanaka Symbols","authors":"Zaifeng Lin,&nbsp;Igor Zelenko","doi":"10.1134/S1560354724020023","DOIUrl":"10.1134/S1560354724020023","url":null,"abstract":"<div><p>The classical result of Eisenhart states that, if a Riemannian metric <span>(g)</span> admits a Riemannian metric that is not constantly proportional to <span>(g)</span> and has the same (parameterized) geodesics as <span>(g)</span> in a neighborhood of a given point, then <span>(g)</span> is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step <span>(2)</span> graded nilpotent Lie algebras, called <span>(mathrm{ad})</span><i>-surjective</i>, and extend the Eisenhart theorem to sub-Riemannian metrics on step <span>(2)</span> distributions with <span>(mathrm{ad})</span>-surjective Tanaka symbols. The class of ad-surjective step <span>(2)</span> nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 2","pages":"304 - 343"},"PeriodicalIF":0.8,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579412","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 Bifurcations of Symmetric Elliptic Orbits 论对称椭圆轨道的分岔
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-03-11 DOI: 10.1134/S1560354724010039
Marina S. Gonchenko
{"title":"On Bifurcations of Symmetric Elliptic Orbits","authors":"Marina S. Gonchenko","doi":"10.1134/S1560354724010039","DOIUrl":"10.1134/S1560354724010039","url":null,"abstract":"<div><p>We study bifurcations of symmetric elliptic fixed points in the case of <i>p</i>:<i>q</i> resonances with odd <span>(qgeqslant 3)</span>. We consider the case where the initial area-preserving map <span>(bar{z}=lambda z+Q(z,z^{*}))</span> possesses the central symmetry, i. e., is invariant under the change of variables <span>(zto-z)</span>, <span>(z^{*}to-z^{*})</span>. We construct normal forms for such maps in the case <span>(lambda=e^{i2pifrac{p}{q}})</span>, where <span>(p)</span> and <span>(q)</span> are mutually prime integer numbers, <span>(pleqslant q)</span> and <span>(q)</span> is odd, and study local bifurcations of the fixed point <span>(z=0)</span> in various settings. We prove the appearance of garlands consisting of four <span>(q)</span>-periodic orbits, two orbits are elliptic and two orbits are saddles, and describe the corresponding bifurcation diagrams for one- and two-parameter families. We also consider the case where the initial map is reversible and find conditions where nonsymmetric periodic orbits of the garlands are nonconservative (contain symmetric pairs of stable and unstable orbits as well as area-contracting and area-expanding saddles).</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 and Dmitry Turaev)","pages":"25 - 39"},"PeriodicalIF":0.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099146","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
Chaos in Coupled Heteroclinic Cycles Between Weak Chimeras 弱嵌合体之间耦合异次元循环中的混沌现象
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-03-11 DOI: 10.1134/S1560354724010131
Artyom E. Emelin, Evgeny A. Grines, Tatiana A. Levanova
{"title":"Chaos in Coupled Heteroclinic Cycles Between Weak Chimeras","authors":"Artyom E. Emelin,&nbsp;Evgeny A. Grines,&nbsp;Tatiana A. Levanova","doi":"10.1134/S1560354724010131","DOIUrl":"10.1134/S1560354724010131","url":null,"abstract":"<div><p>Heteroclinic cycles are widely used in neuroscience in order to mathematically describe different mechanisms of functioning of the brain and nervous system. Heteroclinic cycles and interactions between them can be a source of different types of nontrivial dynamics. For instance, as it was shown earlier, chaotic dynamics can appear as a result of interaction via diffusive couplings between two stable heteroclinic cycles between saddle equilibria. We go beyond these findings by considering two coupled stable heteroclinic cycles rotating in opposite\u0000directions between weak chimeras. Such an ensemble can be mathematically described by a system of six phase equations. Using two-parameter bifurcation analysis, we investigate the scenarios of\u0000emergence and destruction of chaotic dynamics in the system under study.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 and Dmitry Turaev)","pages":"205 - 217"},"PeriodicalIF":0.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099420","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
IN HONOR OF SERGEY GONCHENKO AND VLADIMIR BELYKH 纪念谢尔盖-冈琴科和弗拉基米尔-别列赫
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-03-11 DOI: 10.1134/S1560354724010015
Nikita Barabash, Igor Belykh, Alexey Kazakov, Michael Malkin, Vladimir Nekorkin, Dmitry Turaev
{"title":"IN HONOR OF SERGEY GONCHENKO AND VLADIMIR BELYKH","authors":"Nikita Barabash,&nbsp;Igor Belykh,&nbsp;Alexey Kazakov,&nbsp;Michael Malkin,&nbsp;Vladimir Nekorkin,&nbsp;Dmitry Turaev","doi":"10.1134/S1560354724010015","DOIUrl":"10.1134/S1560354724010015","url":null,"abstract":"","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 and Dmitry Turaev)","pages":"1 - 5"},"PeriodicalIF":0.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140516195","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
Dynamics of a Pendulum in a Rarefied Flow 稀薄流中摆的动力学特性
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-03-11 DOI: 10.1134/S1560354724010088
Alexey Davydov, Alexander Plakhov
{"title":"Dynamics of a Pendulum in a Rarefied Flow","authors":"Alexey Davydov,&nbsp;Alexander Plakhov","doi":"10.1134/S1560354724010088","DOIUrl":"10.1134/S1560354724010088","url":null,"abstract":"<div><p>We consider the dynamics of a rod on the plane in a flow of non-interacting point particles moving at a fixed speed. When colliding with the rod, the particles are reflected elastically and then leave the plane of motion of the rod and do not interact with it. A thin unbending weightless “knitting needle” is fastened to the\u0000massive rod. The needle is attached to an anchor point and can rotate freely about it. The particles do not interact with the needle.</p><p>The equations of dynamics are obtained, which are piecewise analytic: the phase space is divided into four regions where the analytic formulas are different. There are two fixed points of the system, corresponding to the position of the rod parallel to the flow velocity, with the anchor point at the front and the back. It is found that the former point is topologically a stable focus, and the latter is topologically a saddle. A qualitative description of the phase portrait of the system is obtained.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 and Dmitry Turaev)","pages":"134 - 142"},"PeriodicalIF":0.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140098986","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 Study of Discrete Lorenz-Like Attractors 离散类洛伦兹吸引力的数值研究
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-03-11 DOI: 10.1134/S1560354724010064
Alexey Kazakov, Ainoa Murillo, Arturo Vieiro, Kirill Zaichikov
{"title":"Numerical Study of Discrete Lorenz-Like Attractors","authors":"Alexey Kazakov,&nbsp;Ainoa Murillo,&nbsp;Arturo Vieiro,&nbsp;Kirill Zaichikov","doi":"10.1134/S1560354724010064","DOIUrl":"10.1134/S1560354724010064","url":null,"abstract":"<div><p>We consider a homotopic to the identity family of maps, obtained as a discretization of the Lorenz system, such that the dynamics of the last is recovered as a limit dynamics when the discretization parameter tends to zero. We investigate the structure of the discrete Lorenz-like attractors that the map shows for different values of parameters. In particular, we check the pseudohyperbolicity of the observed discrete attractors and show how to\u0000use interpolating vector fields to compute kneading diagrams for near-identity maps. For larger discretization parameter values, the map exhibits what appears to be genuinely-discrete Lorenz-like attractors, that is, discrete chaotic pseudohyperbolic attractors with a negative second Lyapunov exponent. The numerical methods used are general enough to be adapted for arbitrary near-identity discrete systems with similar phase space structure.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 and Dmitry Turaev)","pages":"78 - 99"},"PeriodicalIF":0.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099185","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
Twin Heteroclinic Connections of Reversible Systems 可逆系统的双异次元连接
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-03-11 DOI: 10.1134/S1560354724010040
Nikolay E. Kulagin, Lev M. Lerman, Konstantin N. Trifonov
{"title":"Twin Heteroclinic Connections of Reversible Systems","authors":"Nikolay E. Kulagin,&nbsp;Lev M. Lerman,&nbsp;Konstantin N. Trifonov","doi":"10.1134/S1560354724010040","DOIUrl":"10.1134/S1560354724010040","url":null,"abstract":"<div><p>We examine smooth four-dimensional vector fields reversible under some\u0000smooth involution <span>(L)</span> that has a smooth two-dimensional submanifold of fixed\u0000points. Our main interest here is in the orbit structure of such a system\u0000near two types of heteroclinic connections involving saddle-foci and\u0000heteroclinic orbits connecting them. In both cases we found families of\u0000symmetric periodic orbits, multi-round heteroclinic connections and\u0000countable families of homoclinic orbits of saddle-foci. All this suggests that the orbit\u0000structure near such connections is very complicated. A non-variational version of the stationary Swift – Hohenberg equation is considered, as an example, where such structure has been found numerically.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 and Dmitry Turaev)","pages":"40 - 64"},"PeriodicalIF":0.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099145","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 Homeomorphisms of Three-Dimensional Manifolds with Pseudo-Anosov Attractors and Repellers 论具有伪阿诺索夫吸引子和排斥子的三维流形的同构性
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-03-11 DOI: 10.1134/S1560354724010106
Vyacheslav Z. Grines, Olga V. Pochinka, Ekaterina E. Chilina
{"title":"On Homeomorphisms of Three-Dimensional Manifolds with Pseudo-Anosov Attractors and Repellers","authors":"Vyacheslav Z. Grines,&nbsp;Olga V. Pochinka,&nbsp;Ekaterina E. Chilina","doi":"10.1134/S1560354724010106","DOIUrl":"10.1134/S1560354724010106","url":null,"abstract":"<div><p>The present paper is devoted to a study of orientation-preserving homeomorphisms on three-dimensional manifolds with a non-wandering set consisting of a finite number of surface attractors and repellers. The main results of the paper relate to a class of homeomorphisms for which the restriction of the map to a connected component of the non-wandering set is topologically conjugate to an orientation-preserving pseudo-Anosov homeomorphism. The ambient <span>(Omega)</span>-conjugacy of a homeomorphism from the class with a locally direct product of a pseudo-Anosov homeomorphism and a rough transformation of the circle is proved. In addition, we prove that the centralizer of a pseudo-Anosov homeomorphisms consists of only pseudo-Anosov and periodic maps.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 and Dmitry Turaev)","pages":"156 - 173"},"PeriodicalIF":0.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099200","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 Regularity of Invariant Foliations 论不变叶形的规律性
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-03-11 DOI: 10.1134/S1560354724010027
Dmitry Turaev
{"title":"On the Regularity of Invariant Foliations","authors":"Dmitry Turaev","doi":"10.1134/S1560354724010027","DOIUrl":"10.1134/S1560354724010027","url":null,"abstract":"<div><p>We show that the stable invariant foliation of codimension 1 near a zero-dimensional hyperbolic set of a <span>(C^{beta})</span> map with <span>(beta&gt;1)</span> is <span>(C^{1+varepsilon})</span> with some <span>(varepsilon&gt;0)</span>. The result is applied to the restriction of higher regularity\u0000maps to normally hyperbolic manifolds. An application to the theory of the Newhouse phenomenon is discussed.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 and Dmitry Turaev)","pages":"6 - 24"},"PeriodicalIF":0.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140098869","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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