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Regular and Chaotic Dynamics最新文献

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Sensitivity and Chaoticity of Some Classes of Semigroup Actions 几类半群作用的敏感性和混沌性
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2024-03-11 DOI: 10.1134/S1560354724010118
Nina I. Zhukova
{"title":"Sensitivity and Chaoticity of Some Classes of Semigroup Actions","authors":"Nina I. Zhukova","doi":"10.1134/S1560354724010118","DOIUrl":"10.1134/S1560354724010118","url":null,"abstract":"<div><p>The focus of the work is the investigation of chaos and closely related dynamic properties of continuous actions of almost open\u0000semigroups and <span>(C)</span>-semigroups. The class of dynamical systems <span>((S,X))</span> defined by such semigroups <span>(S)</span> is denoted by <span>(mathfrak{A})</span>.\u0000These semigroups contain, in particular, cascades, semiflows and groups of homeomorphisms. We extend the Devaney definition of chaos to general dynamical systems. For <span>((S,X)inmathfrak{A})</span> on locally compact metric spaces <span>(X)</span> with a countable base we\u0000prove that topological transitivity and density of the set formed by points having closed orbits imply the sensitivity to initial conditions. We assume neither the compactness of metric space nor the compactness of the above-mentioned closed orbits.\u0000In the case when the set of points having compact orbits is dense, our proof proceeds without the assumption of local compactness of the phase space <span>(X)</span>. This statement generalizes the well-known result of J. Banks et al. on Devaney’s definition\u0000of chaos for cascades.The interrelation of sensitivity, transitivity and the property of minimal sets of semigroups is investigated. Various examples are given.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 and Dmitry Turaev)","pages":"174 - 189"},"PeriodicalIF":0.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099201","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
Slow-Fast Systems with an Equilibrium Near the Folded Slow Manifold 在折叠慢速歧面附近达到平衡的慢-快系统
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-12-19 DOI: 10.1134/S156035472354002X
Natalia G. Gelfreikh, Alexey V. Ivanov
{"title":"Slow-Fast Systems with an Equilibrium Near the Folded Slow Manifold","authors":"Natalia G. Gelfreikh,&nbsp;Alexey V. Ivanov","doi":"10.1134/S156035472354002X","DOIUrl":"10.1134/S156035472354002X","url":null,"abstract":"<div><p>We study a slow-fast system with two slow and one fast variables.\u0000We assume that the slow manifold of the system possesses a fold and there is an equilibrium of the system in a small neighborhood of the fold. We derive a normal form for the system\u0000in a neighborhood of the pair “equilibrium-fold”\u0000and study the dynamics of the normal form. In particular, as the ratio of two time scales tends to zero we obtain an asymptotic formula for the Poincaré map\u0000and calculate the parameter values for the first period-doubling bifurcation. The theory is applied to a generalization of the FitzHugh – Nagumo system.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 2","pages":"376 - 403"},"PeriodicalIF":0.8,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138743675","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
Hyperbolic Attractors Which are Anosov Tori 属于阿诺索夫环的双曲吸引子
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-12-19 DOI: 10.1134/S1560354723540018
Marina K. Barinova, Vyacheslav Z. Grines, Olga V. Pochinka, Evgeny V. Zhuzhoma
{"title":"Hyperbolic Attractors Which are Anosov Tori","authors":"Marina K. Barinova,&nbsp;Vyacheslav Z. Grines,&nbsp;Olga V. Pochinka,&nbsp;Evgeny V. Zhuzhoma","doi":"10.1134/S1560354723540018","DOIUrl":"10.1134/S1560354723540018","url":null,"abstract":"<div><p>We consider a topologically mixing hyperbolic attractor <span>(Lambdasubset M^{n})</span> for a diffeomorphism <span>(f:M^{n}to M^{n})</span> of a compact orientable <span>(n)</span>-manifold <span>(M^{n})</span>, <span>(n&gt;3)</span>. Such an attractor <span>(Lambda)</span> is called an Anosov torus provided the restriction <span>(f|_{Lambda})</span> is conjugate to Anosov algebraic automorphism of <span>(k)</span>-dimensional torus <span>(mathbb{T}^{k})</span>.\u0000We prove that <span>(Lambda)</span> is an Anosov torus for two cases:\u00001) <span>(dim{Lambda}=n-1)</span>, <span>(dim{W^{u}_{x}}=1)</span>, <span>(xinLambda)</span>;\u00002) <span>(dimLambda=k,dim W^{u}_{x}=k-1,xinLambda)</span>, and <span>(Lambda)</span> belongs to an <span>(f)</span>-invariant closed <span>(k)</span>-manifold, <span>(2leqslant kleqslant n)</span>, topologically embedded in <span>(M^{n})</span>.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 2","pages":"369 - 375"},"PeriodicalIF":0.8,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138743744","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 and Hyperchaos in Two Coupled Identical Hindmarsh – Rose Systems 两个完全相同的辛德马什-罗斯耦合系统中的混沌与超混沌
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-12-19 DOI: 10.1134/S1560354723540031
Nataliya V. Stankevich, Andrey A. Bobrovskii, Natalya A. Shchegoleva
{"title":"Chaos and Hyperchaos in Two Coupled Identical Hindmarsh – Rose Systems","authors":"Nataliya V. Stankevich,&nbsp;Andrey A. Bobrovskii,&nbsp;Natalya A. Shchegoleva","doi":"10.1134/S1560354723540031","DOIUrl":"10.1134/S1560354723540031","url":null,"abstract":"<div><p>The dynamics of two coupled neuron models, the Hindmarsh – Rose systems, are studied. Their\u0000interaction is simulated via a chemical coupling that is implemented with a sigmoid function.\u0000It is shown that the model may exhibit complex behavior: quasi-periodic, chaotic and\u0000hyperchaotic oscillations. A phenomenological scenario for the formation of hyperchaos\u0000associated with the appearance of a discrete Shilnikov attractor is described. It is shown\u0000that the formation of these attractors leads to the appearance of in-phase bursting\u0000oscillations.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 and Dmitry Turaev)","pages":"120 - 133"},"PeriodicalIF":0.8,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138745924","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
Non-Quasi-Periodic Normal Form Theory 非准周期正态理论
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-12-07 DOI: 10.1134/S1560354723060035
Gabriella Pinzari
{"title":"Non-Quasi-Periodic Normal Form Theory","authors":"Gabriella Pinzari","doi":"10.1134/S1560354723060035","DOIUrl":"10.1134/S1560354723060035","url":null,"abstract":"<div><p>We review a recent application of the ideas of normal form theory to systems (Hamiltonian ones or general ODEs) where the perturbing term is not periodic in one coordinate variable. The main difference\u0000from the standard case consists in the non-uniqueness of the normal form and the total absence of the small\u0000divisors problem. The exposition is quite general, so as to allow extensions to the case\u0000of more non-periodic coordinates, and more functional settings. Here, for simplicity,\u0000we work in the real-analytic class.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 6","pages":"841 - 864"},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555304","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
Unifying the Hyperbolic and Spherical (2)-Body Problem with Biquaternions 用双四元数统一双曲和球面 $$2$ 天体问题
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-12-07 DOI: 10.1134/S1560354723060011
Philip Arathoon
{"title":"Unifying the Hyperbolic and Spherical (2)-Body Problem with Biquaternions","authors":"Philip Arathoon","doi":"10.1134/S1560354723060011","DOIUrl":"10.1134/S1560354723060011","url":null,"abstract":"<div><p>The <span>(2)</span>-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexification of a spherical system. In this way, results for the <span>(2)</span>-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the <span>(2)</span>-body problem on hyperbolic 3-space and discuss their stability for a strictly attractive potential.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 6","pages":"822 - 834"},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555416","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 Method of Introduction of Local Variables in a Neighborhood of Periodic Solution of a Hamiltonian System with Two Degrees of Freedom 论在具有两个自由度的哈密尔顿系统的周期解邻域中引入局部变量的方法
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2023-12-07 DOI: 10.1134/S1560354723060059
Boris S. Bardin
{"title":"On the Method of Introduction of Local Variables in a Neighborhood of Periodic Solution of a Hamiltonian System with Two Degrees of Freedom","authors":"Boris S. Bardin","doi":"10.1134/S1560354723060059","DOIUrl":"10.1134/S1560354723060059","url":null,"abstract":"<div><p>A general method is presented for constructing a nonlinear canonical transformation, which makes it possible to introduce local variables in a neighborhood of periodic motions of an autonomous Hamiltonian system with two degrees of freedom. This method can be used for investigating the behavior of the Hamiltonian system in\u0000the vicinity of its periodic trajectories. In particular, it can be applied to solve the problem of orbital stability of periodic motions.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 6","pages":"878 - 887"},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138563720","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
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
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