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Regular and Chaotic Dynamics最新文献

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Sonya Kowalewski’s Legacy to Mechanics and Complex Lie Algebras Sonya Kowalewski对力学和复李代数的遗产
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2025-10-10 DOI: 10.1134/S1560354725050028
Velimir Jurdjevic
{"title":"Sonya Kowalewski’s Legacy to Mechanics and Complex Lie Algebras","authors":"Velimir Jurdjevic","doi":"10.1134/S1560354725050028","DOIUrl":"10.1134/S1560354725050028","url":null,"abstract":"<div><p>This paper provides an original rendition of the heavy top that unravels the mysteries behind S. Kowalewski’s seminal work on the motions of a rigid body around a fixed point under the influence of gravity.\u0000The point of departure for understanding Kowalewski’s work\u0000begins with Kirchhoff’s model for the equilibrium configurations of an elastic rod in <span>({mathbb{R}}^{3})</span> subject to fixed bending and twisting moments at its ends [17]. This initial orientation to the elastic problem shows, first, that the Kowalewski type integrals discovered by I. V. Komarov and V. B. Kuznetsov [24, 25] appear naturally on the Lie algebras associated with the orthonormal frame bundles of the sphere <span>(S^{3})</span> and the hyperboloid <span>(H^{3})</span> [17] and, secondly, it shows\u0000that these integrals of motion can be naturally extracted from a canonical Poisson system on the dual of <span>(so(4,mathbb{C}))</span> generated by\u0000an affine quadratic Hamiltonian <span>(H)</span> (Kirchhoff – Kowalewski type).</p><p>The paper shows that the passage to complex variables\u0000is synonymous with the representation of <span>(so(4,mathbb{C}))</span> as <span>(sl(2,mathbb{C})times sl(2,mathbb{C}))</span> and the embedding of <span>(H)</span> into <span>(sp(4,mathbb{C}))</span>, an important intermediate step towards uncovering the origins of Kowalewski’s integral. There is a quintessential Kowalewski type integral of motion on <span>(sp(4,mathbb{C}))</span> that appears as a spectral invariant for the Poisson system associated with a Hamiltonian <span>(mathcal{H})</span> (a natural extension of <span>(H)</span>) that satisfies Kowalewski’s conditions.</p><p>The text then demonstrates the relevance of this integral of motion for other studies in the existing literature [7, 35]. The text also includes a self-contained treatment of the integration of the Kowalewski type equations based on Kowalewski’s ingenuous separation of variables, the hyperelliptic curve and the solutions on its Jacobian variety.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"767 - 798"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256372","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
Painlevé Test, First Integrals and Exact Solutions of Nonlinear Dissipative Differential Equations 非线性耗散微分方程的疼痛水平检验、第一积分和精确解
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2025-10-10 DOI: 10.1134/S1560354725050041
Nikolay A. Kudryashov
{"title":"Painlevé Test, First Integrals and Exact Solutions of Nonlinear Dissipative Differential Equations","authors":"Nikolay A. Kudryashov","doi":"10.1134/S1560354725050041","DOIUrl":"10.1134/S1560354725050041","url":null,"abstract":"<div><p>The Korteweg – de Vries – Burgers equation, the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation and the nonlinear differential equation for description surface waves in a convecting fluid are considered. The Cauchy problems for all these partial differential equations are not solved by the\u0000inverse scattering transform. Reductions of these equations to nonlinear ordinary differential\u0000equations do not pass the Painlevé test. However, there are local expansions of the general\u0000solutions in the Laurent series near movable singular points.\u0000We demonstrate that the obtained information from the Painlevé test for reductions of\u0000nonlinear evolution dissipative differential equations can be used to construct the\u0000nonautonomous first integrals of nonlinear ordinary differential equations. Taking into\u0000account the found first integrals, we also obtain analytical solutions of nonlinear evolution\u0000dissipative differential equations. Our approach is illustrated to obtain the\u0000nonautonomous first integrals for reduction of the Korteweg – de Vries – Burgers equation,\u0000the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation and\u0000the nonlinear differential equation for description surface waves in a convecting fluid.\u0000The obtained first integrals are used to construct exact solutions of the above-mentioned\u0000nonlinear evolution equations with as many arbitrary constants as possible. It is shown that\u0000some exact solutions of the equation for description of nonlinear waves in a convecting\u0000liquid are expressed via the Painlevé transcendents.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"819 - 836"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256268","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
Periodic Response Solutions to Multidimensional Nonlinear Schrödinger Equations with Unbounded Perturbation 具有无界摄动的多维非线性Schrödinger方程的周期响应解
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2025-09-25 DOI: 10.1134/S1560354725530048
Zuhong You, Xiaoping Yuan
{"title":"Periodic Response Solutions to Multidimensional Nonlinear Schrödinger Equations with Unbounded Perturbation","authors":"Zuhong You,&nbsp;Xiaoping Yuan","doi":"10.1134/S1560354725530048","DOIUrl":"10.1134/S1560354725530048","url":null,"abstract":"<div><p>This paper establishes the existence of periodic response solutions for multi-dimensional nonlinear Schrödinger equations (NLS) with unbounded perturbations. We adapt the Craig – Wayne – Bourgain (CWB) method to this context. The central point is that the core separation lemma remains valid for sufficiently weak unbounded perturbations, which allows the CWB method to be successfully applied. This work extends the applicability of the CWB method to problems with unbounded perturbations.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 3","pages":"438 - 452"},"PeriodicalIF":0.8,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148060748","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
Computation of Periodic Libration Point Orbits in the Circular Restricted Three-Body Problem 圆形受限三体问题中周期振动点轨道的计算
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2025-09-25 DOI: 10.1134/S1560354725530024
Daniella Gafurova, Sergey Aksenov
{"title":"Computation of Periodic Libration Point Orbits in the Circular Restricted Three-Body Problem","authors":"Daniella Gafurova,&nbsp;Sergey Aksenov","doi":"10.1134/S1560354725530024","DOIUrl":"10.1134/S1560354725530024","url":null,"abstract":"<div><p>The circular restricted three-body problem is used as an approximate model in space mission planning. Its periodic solutions around equilibrium points, which are referred to as the libration points, are utilized for exploration of possible spacecraft trajectories in the preliminary stages of mission design. In this paper, a numerical methodology for periodic libration point orbits (LPOs) computation is introduced and applied to the construction and study of <i>N</i>-periodic (up to <span>(N=6)</span>) quasi-planar orbit families in the Earth-Moon system. The stability and the bifurcation points of these families are determined. The proposed method is based on an iterative algorithm searching for the initial state vector of periodic LPOs, which allows computing unstable long-periodic and large-amplitude orbits. The method is suited to perform a straightforward switch to bifurcating branches of periodic orbits.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"969 - 991"},"PeriodicalIF":0.8,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145646213","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 Stability of a Charged Pendulum with an Oscillating Suspension Point Located above Two Electric Charges of the Same Intensity and Sign,Equally Distant from the Suspension Point 振荡悬点位于两个强度和符号相同、离悬点距离相等的电荷之上的带电摆的参数稳定性
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2025-09-25 DOI: 10.1134/S1560354725530012
Gerson C. Araujo, Keity Murielly J. de Andrade, José Laudelino de Menezes Neto
{"title":"Parametric Stability of a Charged Pendulum with an Oscillating Suspension Point Located above Two Electric Charges of the Same Intensity and Sign,\u0000Equally Distant from the Suspension Point","authors":"Gerson C. Araujo,&nbsp;Keity Murielly J. de Andrade,&nbsp;José Laudelino de Menezes Neto","doi":"10.1134/S1560354725530012","DOIUrl":"10.1134/S1560354725530012","url":null,"abstract":"<div><p>In this study, we analyze a planar mathematical pendulum whose suspension point oscillates vertically according to a harmonic law. The pendulum bob is electrically charged and positioned slightly above two electric charges of equal sign and intensity, which are equidistant from the suspension point and separated by a distance of <span>(2d.)</span> Here, <span>(d)</span> denotes the distance from each charge to the orthogonal projection of the suspension point onto the horizontal line where the charges lie. We formulate the Hamiltonian structure of this mechanical system, identify two equilibrium points, and examine the system’s linear stability. The dynamics are governed by three dimensionless\u0000parameters: <span>(mu)</span> which relates to the electric charges; <span>(varepsilon,)</span> associated with the amplitude of oscillation of the suspension point; and <span>(alpha,)</span> determined by the frequency of the system. We then investigate the parametric stability of the equilibrium points. Finally, we present the boundary surfaces that separate regions of stability and instability in the parameter space. For specific values of <span>(mu,)</span> we derive cross-sectional curves that delineate these regions, using results from the Krein – Gelfand – Lidskii theorem and the Deprit – Hori\u0000method.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 3","pages":"484 - 501"},"PeriodicalIF":0.8,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148060750","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 Construction of Solutions of the Davey – Stewartson I Equation via Dressing Chain 用修整链构造Davey - Stewartson I方程的解
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2025-09-25 DOI: 10.1134/S1560354725530036
Ismagil T. Habibullin, Aigul R. Khakimova
{"title":"On the Construction of Solutions of the Davey – Stewartson I Equation via Dressing Chain","authors":"Ismagil T. Habibullin,&nbsp;Aigul R. Khakimova","doi":"10.1134/S1560354725530036","DOIUrl":"10.1134/S1560354725530036","url":null,"abstract":"<div><p>An original effective method for constructing explicit solutions of integrable Davey – Stewartson type equations is proposed, based on the use of dressing chains. The main difficulty arising when using the symmetry approach in 3D is associated with nonlocal variables entering the equation. To solve the nonlocality problem, it is proposed to replace the infinite dressing chain with its finite-field reductions preserving the integrability property. The application of the method is illustrated by the DS I equation, for which a new class of explicit solutions is constructed that depend on two arbitrary functions. In this example, the dressing chain is replaced by a finite-field reduction of the Toda lattice corresponding to a simple Lie algebra <span>(A_{2})</span>.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 3","pages":"426 - 437"},"PeriodicalIF":0.8,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148060749","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
Integrability of Homogeneous Exact Magnetic Flows on Spheres 球上均匀精确磁流的可积性
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2025-08-11 DOI: 10.1134/S1560354725040082
Vladimir Dragović, Borislav Gajić, Božidar Jovanović
{"title":"Integrability of Homogeneous Exact Magnetic Flows on Spheres","authors":"Vladimir Dragović,&nbsp;Borislav Gajić,&nbsp;Božidar Jovanović","doi":"10.1134/S1560354725040082","DOIUrl":"10.1134/S1560354725040082","url":null,"abstract":"<div><p>We consider motion of a material point placed in a constant homogeneous magnetic field in <span>(mathbb{R}^{n})</span> and also motion restricted to the sphere <span>(S^{n-1})</span>.\u0000While there is an obvious integrability of the magnetic system in <span>(mathbb{R}^{n})</span>, the integrability of the system restricted to the sphere <span>(S^{n-1})</span> is highly nontrivial. We prove\u0000complete integrability of the obtained restricted magnetic systems for <span>(nleqslant 6)</span>. The first integrals of motion of the magnetic flows on the spheres <span>(S^{n-1})</span>, for <span>(n=5)</span> and <span>(n=6)</span>, are polynomials of degree\u0000<span>(1)</span>, <span>(2)</span>, and <span>(3)</span> in momenta.\u0000We prove noncommutative integrability of the obtained magnetic flows for any <span>(ngeqslant 7)</span> when the systems allow a reduction to the cases with <span>(nleqslant 6)</span>. We conjecture that the restricted magnetic systems on <span>(S^{n-1})</span> are integrable for all <span>(n)</span>.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"582 - 597"},"PeriodicalIF":0.8,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121783","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 Oscillations in a Neighborhood of Lagrangian Libration Points in OneResonance Case 单共振情况下拉格朗日振动点邻域内的振动
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2025-08-11 DOI: 10.1134/S1560354725040136
Anatoly P. Markeev
{"title":"On Oscillations in a Neighborhood of Lagrangian Libration Points in One\u0000Resonance Case","authors":"Anatoly P. Markeev","doi":"10.1134/S1560354725040136","DOIUrl":"10.1134/S1560354725040136","url":null,"abstract":"<div><p>This paper addresses the spatial restricted elliptic problem\u0000of three bodies (material points) gravitating toward each other under Newton’s law of\u0000gravitation. The eccentricity of the orbit of the main attracting bodies is assumed to be\u0000small, and nonlinear oscillations of\u0000a passively gravitating body near a Lagrangian triangular libration point are studied.\u0000It is assumed that in the limiting case of the circular problem the ratio\u0000of the frequency of rotation of the main bodies about their common center of mass\u0000to the value of one of the frequencies of small linear oscillations of the passive body\u0000is exactly equal to three. A detailed analysis is made of two different particular cases of\u0000influence of the three-dimensionality of the\u0000problem on the characteristics of nonlinear oscillations of the passive body.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"666 - 676"},"PeriodicalIF":0.8,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121789","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
Spinning Top in Quadratic Potential and Matrix Dressing Chain 二次势和矩阵修整链中的纺丝顶
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2025-08-11 DOI: 10.1134/S1560354725040021
Vsevolod E. Adler, Alexander P. Veselov
{"title":"Spinning Top in Quadratic Potential and Matrix Dressing Chain","authors":"Vsevolod E. Adler,&nbsp;Alexander P. Veselov","doi":"10.1134/S1560354725040021","DOIUrl":"10.1134/S1560354725040021","url":null,"abstract":"<div><p>We show that the equations of motion of a rigid body about a fixed point in the Newtonian\u0000field with a quadratic potential are special reduction of period-one closure of the Darboux dressing chain for the Schrödinger operators with matrix potentials. Some new explicit solutions of the corresponding matrix system and the spectral properties of the related Schrödinger operators are discussed.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"464 - 480"},"PeriodicalIF":0.8,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121790","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
Lyapunov Exponents of Linear Switched Systems 线性切换系统的Lyapunov指数
IF 0.8 4区 数学
Regular and Chaotic Dynamics Pub Date : 2025-08-11 DOI: 10.1134/S1560354725040033
Andrei A. Agrachev, Michele Motta
{"title":"Lyapunov Exponents of Linear Switched Systems","authors":"Andrei A. Agrachev,&nbsp;Michele Motta","doi":"10.1134/S1560354725040033","DOIUrl":"10.1134/S1560354725040033","url":null,"abstract":"<div><p>We explicitly compute the maximal Lyapunov exponent for a switched system on <span>(mathrm{SL}_{2}(mathbb{R}))</span> and the corresponding switching function which realizes the maximal exponent. This computation is reduced to the characterization of optimal trajectories for an optimal control problem on the Lie group.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"481 - 503"},"PeriodicalIF":0.8,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121373","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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