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Circular Fleitas Scheme for Gradient-Like Flows on the Surface 表面梯度流的循环弗莱塔方案
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-12-07 DOI: 10.1134/S1560354723060047
Vladislav D. Galkin, Elena V. Nozdrinova, Olga V. Pochinka
{"title":"Circular Fleitas Scheme for Gradient-Like Flows on the Surface","authors":"Vladislav D. Galkin,&nbsp;Elena V. Nozdrinova,&nbsp;Olga V. Pochinka","doi":"10.1134/S1560354723060047","DOIUrl":"10.1134/S1560354723060047","url":null,"abstract":"<div><p>In this paper, we obtain a classification of gradient-like\u0000flows on arbitrary surfaces by generalizing the circular\u0000Fleitas\u0000scheme. In 1975 he proved that such a scheme is a complete\u0000invariant of topological equivalence for polar flows on 2- and 3-manifolds.\u0000In this paper, we generalize the concept of a circular scheme\u0000to arbitrary gradient-like flows on surfaces. We prove that the\u0000isomorphism class of such schemes is a complete invariant of\u0000topological equivalence. We also solve exhaustively the\u0000realization problem by describing an abstract circular\u0000scheme and the process of realizing a gradient-like flow on\u0000the surface. In addition, we construct an efficient algorithm\u0000for distinguishing the isomorphism of circular schemes.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 6","pages":"865 - 877"},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555422","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
Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base Using Feedback 利用反馈稳定振动底座上球形机器人的稳定旋转
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-12-07 DOI: 10.1134/S1560354723060060
Alexander A. Kilin, Tatiana B. Ivanova, Elena N. Pivovarova
{"title":"Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base Using Feedback","authors":"Alexander A. Kilin,&nbsp;Tatiana B. Ivanova,&nbsp;Elena N. Pivovarova","doi":"10.1134/S1560354723060060","DOIUrl":"10.1134/S1560354723060060","url":null,"abstract":"<div><p>This paper treats the problem of a spherical robot with an axisymmetric pendulum drive\u0000rolling without slipping on a vibrating plane. The main purpose of the paper is\u0000to investigate the stabilization of the upper vertical rotations of the pendulum\u0000using feedback (additional control action). For the chosen type of feedback,\u0000regions of asymptotic stability of the upper vertical rotations of the pendulum are constructed\u0000and possible bifurcations are analyzed. Special attention is also given to the question of\u0000the stability of periodic solutions arising as the vertical rotations lose stability.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 6","pages":"888 - 905"},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1134/S1560354723060060.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138563353","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
Non-Integrable Sub-Riemannian Geodesic Flow on (J^{2}(mathbb{R}^{2},mathbb{R})) $$J^{2}(mathbb{R}^{2},mathbb{R})$$上的非不可测次黎曼大地流
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-12-07 DOI: 10.1134/S1560354723060023
Alejandro Bravo-Doddoli
{"title":"Non-Integrable Sub-Riemannian Geodesic Flow on (J^{2}(mathbb{R}^{2},mathbb{R}))","authors":"Alejandro Bravo-Doddoli","doi":"10.1134/S1560354723060023","DOIUrl":"10.1134/S1560354723060023","url":null,"abstract":"<div><p>The space of <span>(2)</span>-jets of a real function of two real variables, denoted by <span>(J^{2}(mathbb{R}^{2},mathbb{R}))</span>, admits the structure of a metabelian Carnot group, so <span>(J^{2}(mathbb{R}^{2},mathbb{R}))</span> has a normal abelian sub-group <span>(mathbb{A})</span>. As any sub-Riemannian manifold, <span>(J^{2}(mathbb{R}^{2},mathbb{R}))</span> has an associated Hamiltonian geodesic flow. The Hamiltonian action of <span>(mathbb{A})</span> on <span>(T^{*}J^{2}(mathbb{R}^{2},mathbb{R}))</span> yields the reduced Hamiltonian <span>(H_{mu})</span> on <span>(T^{*}mathcal{H}simeq T^{*}(J^{2}(mathbb{R}^{2},mathbb{R})/mathbb{A}))</span>, where <span>(H_{mu})</span> is a two-dimensional Euclidean space. The paper is devoted to proving that the reduced Hamiltonian <span>(H_{mu})</span> is non-integrable by meromorphic functions for some values of <span>(mu)</span>. This result suggests the sub-Riemannian geodesic flow on <span>(J^{2}(mathbb{R}^{2},mathbb{R}))</span> is not meromorphically integrable.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 6","pages":"835 - 840"},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555578","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
Side-Comparison for Transition Maps in Multi-Layer Canard Problems 多层Canard问题中过渡映射的边比较
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-10-20 DOI: 10.1134/S1560354723040159
Peter De Maesschalck, Freddy Dumortier, Robert Roussarie
{"title":"Side-Comparison for Transition Maps in Multi-Layer Canard Problems","authors":"Peter De Maesschalck,&nbsp;Freddy Dumortier,&nbsp;Robert Roussarie","doi":"10.1134/S1560354723040159","DOIUrl":"10.1134/S1560354723040159","url":null,"abstract":"<div><p>The paper deals with multi-layer canard cycles, extending the results of [1]. As a practical tool we introduce the connection diagram of a canard cycle and we show how to determine it in an easy way. This connection diagram presents in a clear way all available information that is necessary to formulate the main system of equations used in the study of the bifurcating limit cycles. In a forthcoming paper we will show that both the type of the layers and the nature of the connections between the layers play an essential role in determining the number and the bifurcations of the limit cycles that can be created from a canard cycle.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"763 - 780"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50500796","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
Three-Body Relative Equilibria on (mathbb{S}^{2}) 关于（mathbb｛S｝^｛2｝）的三体相对平衡
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-10-20 DOI: 10.1134/S1560354723040111
Toshiaki Fujiwara, Ernesto Pérez-Chavela
{"title":"Three-Body Relative Equilibria on (mathbb{S}^{2})","authors":"Toshiaki Fujiwara,&nbsp;Ernesto Pérez-Chavela","doi":"10.1134/S1560354723040111","DOIUrl":"10.1134/S1560354723040111","url":null,"abstract":"<div><p>We study relative equilibria (<span>(RE)</span>) for the three-body problem\u0000on <span>(mathbb{S}^{2})</span>,\u0000under the influence of a general potential which only depends on\u0000<span>(cossigma_{ij})</span> where <span>(sigma_{ij})</span> are the mutual angles\u0000among the masses.\u0000Explicit conditions for\u0000masses <span>(m_{k})</span> and <span>(cossigma_{ij})</span>\u0000to form relative equilibrium are shown.\u0000Using the above conditions,\u0000we study the equal masses case\u0000under the cotangent potential.\u0000We show the existence of\u0000scalene, isosceles, and equilateral Euler <span>(RE)</span>, and isosceles\u0000and equilateral Lagrange <span>(RE)</span>.\u0000We also show that\u0000the equilateral Euler <span>(RE)</span> on a rotating meridian\u0000exists for general potential <span>(sum_{i&lt;j}m_{i}m_{j}U(cossigma_{ij}))</span>\u0000with any mass ratios.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"690 - 706"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50500792","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 Phase at a Resonance in Slow-Fast Hamiltonian Systems 关于慢-快哈密顿系统共振时的相位
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-10-20 DOI: 10.1134/S1560354723040068
Yuyang Gao, Anatoly Neishtadt, Alexey Okunev
{"title":"On Phase at a Resonance in Slow-Fast Hamiltonian Systems","authors":"Yuyang Gao,&nbsp;Anatoly Neishtadt,&nbsp;Alexey Okunev","doi":"10.1134/S1560354723040068","DOIUrl":"10.1134/S1560354723040068","url":null,"abstract":"<div><p>We consider a slow-fast Hamiltonian system with one fast angle variable (a fast phase) whose frequency vanishes on some surface in the space of slow variables (a resonant surface). Systems of such form appear in the study of dynamics of charged particles in an inhomogeneous magnetic field\u0000under the influence of high-frequency electrostatic waves. Trajectories of the system averaged over the fast phase cross the resonant surface.\u0000The fast phase makes <span>(simfrac{1}{varepsilon})</span> turns before arrival at the resonant surface (<span>(varepsilon)</span> is a small parameter of the problem). An asymptotic formula for the value of the phase at the arrival at the resonance\u0000was derived earlier in the context of study of charged particle dynamics on the basis of heuristic\u0000considerations without any estimates of its accuracy. We provide a rigorous derivation of this formula and prove that its accuracy is <span>(O(sqrt{varepsilon}))</span> (up to a logarithmic correction). This estimate for the accuracy is optimal.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"585 - 612"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50500789","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
To Alain Chenciner On his 80th Birthday 致阿兰·钦奇纳80岁生日
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-10-20 DOI: 10.1134/S1560354723040019
{"title":"To Alain Chenciner On his 80th Birthday","authors":"","doi":"10.1134/S1560354723040019","DOIUrl":"10.1134/S1560354723040019","url":null,"abstract":"","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"333 - 342"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50500788","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
Polynomial Entropy and Polynomial Torsion for Fibered Systems 纤维系统的多项式熵和多项式扭转
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-10-20 DOI: 10.1134/S156035472304007X
Flavien Grycan-Gérard, Jean-Pierre Marco
{"title":"Polynomial Entropy and Polynomial Torsion for Fibered Systems","authors":"Flavien Grycan-Gérard,&nbsp;Jean-Pierre Marco","doi":"10.1134/S156035472304007X","DOIUrl":"10.1134/S156035472304007X","url":null,"abstract":"<div><p>Given a continuous fibered dynamical system, we first introduce the notion of polynomial torsion of a fiber,\u0000which measures the “infinitesimal variation” of the dynamics between the fiber and the neighboring ones.\u0000This gives rise to an (upper semicontinous) torsion function,\u0000defined on the base of the system, which is a new\u0000<span>(C^{0})</span> (fiber) conjugacy invariant. We prove that the polynomial entropy of the system is the supremum of\u0000the torsion of its fibers, which yields a new insight into the creation of polynomial entropy in fibered systems.\u0000We examine the relevance of these results in the context of integrable Hamiltonian\u0000systems or diffeomorphisms, with the particular cases of <span>(C^{0})</span>-integrable twist maps on the annulus and geodesic flows.\u0000Finally, we bound from below the polynomial entropy of <span>(ell)</span>-modal interval maps in terms of their lap number and answer a question by Gomes and Carneiro.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"613 - 627"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1134/S156035472304007X.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50500790","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
Compactification of the Energy Surfaces for (n) Bodies 物体能量表面的紧致化
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-10-20 DOI: 10.1134/S1560354723040081
Andreas Knauf, Richard Montgomery
{"title":"Compactification of the Energy Surfaces for (n) Bodies","authors":"Andreas Knauf,&nbsp;Richard Montgomery","doi":"10.1134/S1560354723040081","DOIUrl":"10.1134/S1560354723040081","url":null,"abstract":"<div><p>For <span>(n)</span> bodies moving in Euclidean <span>(d)</span>-space under the influence of a\u0000homogeneous pair interaction we\u0000compactify every center of mass energy surface, obtaining a\u0000<span>(big{(}2d(n-1)-1big{)})</span>-dimensional manifold with corners in the sense of Melrose.\u0000After a time change, the flow on this manifold is globally defined\u0000and nontrivial on the boundary.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"628 - 658"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50500791","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
A Simple Proof of Gevrey Estimates for Expansions of Quasi-Periodic Orbits: Dissipative Models and Lower-Dimensional Tori 拟周期轨道展开的Gevrey估计的一个简单证明：耗散模型和低维Tori
IF 1.4 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-10-20 DOI: 10.1134/S1560354723040123
Adrián P. Bustamante, Rafael de la Llave
{"title":"A Simple Proof of Gevrey Estimates for Expansions of Quasi-Periodic Orbits: Dissipative Models and Lower-Dimensional Tori","authors":"Adrián P. Bustamante,&nbsp;Rafael de la Llave","doi":"10.1134/S1560354723040123","DOIUrl":"10.1134/S1560354723040123","url":null,"abstract":"<div><p>We consider standard-like/Froeschlé dissipative maps\u0000with a dissipation and nonlinear perturbation. That is,\u0000</p>\u0000 <div><div><span>\u0000$$T_{varepsilon}(p,q)=left((1-gammavarepsilon^{3})p+mu+varepsilon V^{prime}(q),q+(1-gammavarepsilon^{3})p+mu+varepsilon V^{prime}(q)bmod 2piright)$$\u0000</span></div></div>\u0000 <p>\u0000where <span>(pin{mathbb{R}}^{D})</span>, <span>(qin{mathbb{T}}^{D})</span> are the dynamical\u0000variables. We fix a frequency <span>(omegain{mathbb{R}}^{D})</span> and study the existence of\u0000quasi-periodic orbits. When there is dissipation, having\u0000a quasi-periodic orbit of frequency <span>(omega)</span> requires\u0000selecting the parameter <span>(mu)</span>, called <i>the drift</i>.</p><p>We first study the Lindstedt series (formal power series in <span>(varepsilon)</span>) for quasi-periodic orbits with <span>(D)</span> independent frequencies and the drift when <span>(gammaneq 0)</span>.\u0000We show that, when <span>(omega)</span> is\u0000irrational, the series exist to all orders, and when <span>(omega)</span> is Diophantine,\u0000we show that the formal Lindstedt series are Gevrey.\u0000The Gevrey nature of the Lindstedt series above was shown\u0000in [3] using a more general method, but the present proof is\u0000rather elementary.</p><p>We also study the case when <span>(D=2)</span>, but the quasi-periodic orbits\u0000have only one independent frequency (lower-dimensional tori).\u0000Both when <span>(gamma=0)</span> and when <span>(gammaneq 0)</span>, we show\u0000that, under some mild nondegeneracy conditions on <span>(V)</span>, there\u0000are (at least two) formal Lindstedt series defined to all orders\u0000and that they are Gevrey.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"707 - 730"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50500793","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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