Regular and Chaotic Dynamics最新文献

{"title":"Sigma Function Associated with a Hyperelliptic Curve with Two Points at Infinity","authors":"Takanori Ayano,&nbsp;Victor M. Buchstaber","doi":"10.1134/S1560354725540020","DOIUrl":"10.1134/S1560354725540020","url":null,"abstract":"<div><p>Baker constructed basic meromorphic functions on the Jacobian variety of a hyperelliptic curve with two points at infinity.\u0000We call them Baker functions.\u0000The construction is based on the Abel – Jacobi map, which allows us to identify the field of meromorphic functions on the Jacobian variety of the curve with the field of meromorphic functions on the symmetric product of the curve.\u0000In our previous paper, a solution to the KP equation was constructed in terms of the Baker function.\u0000This paper is devoted to the properties of the Baker functions.\u0000In this paper, we construct an entire function whose second logarithmic derivatives are the Baker functions.\u0000We prove that the power series expansion of the entire function around the origin is determined only by the coefficients of the defining equation of the curve and a branch point of the curve algebraically.\u0000We also describe the quasi-periodicity of the entire function and express the entire function in terms of the Riemann theta function.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 Editors:","pages":"69 - 82"},"PeriodicalIF":0.8,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147340628","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}

{"title":"On Perturbations of the Spectrum of\u0000a One-Dimensional (mathcal{PT})-Symmetric Periodic Schrödinger Operator","authors":"Petr G. Grinevich,&nbsp;Iskander A. Taimanov","doi":"10.1134/S1560354725060036","DOIUrl":"10.1134/S1560354725060036","url":null,"abstract":"<div><p>For a <span>(mathcal{PT})</span>-symmetric periodic Schrödinger operator, which is a small perturbation of the zero potential, we calculate the spectrum and the divisor of zeroes of the Bloch function in the leading order of perturbation theory. In particular, we show that the analogs of lacunae of the Bloch spectrum are ellipses, and their focal points coincide with the branch points of the spectral curve.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"962 - 968"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145646145","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}

{"title":"The Lagrange Identity and Dynamics in a Potential Jacobi Field","authors":"Valery V. Kozlov","doi":"10.1134/S1560354725060012","DOIUrl":"10.1134/S1560354725060012","url":null,"abstract":"<div><p>A Jacobi field is a potential force field whose potential is a homogeneous function of degree\u0000<span>(-2)</span>. The problem of the motion of a particle in such a field admits an additional integral\u0000quadratic in velocities. It can be used to reduce the number of degrees of freedom and to\u0000pass to the study of a reduced system with spherical configuration space. These results\u0000are extended to the more general case of the motion of a particle\u0000in spaces of constant curvature. An analysis is made of particle motion\u0000on a cone whose vertex coincides with the singular point of the\u0000Jacobi potential. A lower estimate of the distance from the moving particle to the vertex of the cone is given. This approach is also applicable to a more general case where the charged particle\u0000is additionally located in the magnetic field of a monopole. A billiard inside the cone\u0000with a particle bouncing elastically off its boundary is\u0000considered.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"887 - 907"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145646266","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}

{"title":"New Dynamical Mechanisms of Quenching in a System of Coupled Bautin Oscillators","authors":"Andrey A. Markelov,&nbsp;Aleksey S. Dmitrichev,&nbsp;Vladimir I. Nekorkin","doi":"10.1134/S1560354725060048","DOIUrl":"10.1134/S1560354725060048","url":null,"abstract":"<div><p>A system of two diffusively coupled Bautin (generalized Stuart – Landau) oscillators is considered. Using a specially designed reduced system, the existence and stability of homogeneous solutions are investigated. Such solutions represent oscillatory regimes in which the amplitudes of different oscillators are identical to each other and coincide at any given time. A partition of “coupling strength — frequency mismatch” parameter plane into regions with different dynamical behavior of the oscillators is obtained. It is established that the phase space of the system has a foliation into a continuum of two-dimensional invariant manifolds. It is shown that oscillation quenching in the system, in contrast to systems of diffusively coupled Stuart – Landau oscillators, is determined by new mechanisms and is associated with the bifurcation of merger of invariant tori and the saddle-node (tangent) bifurcations of limit cycles. At the same time, the quenching does not occur monotonously with a change in the coupling strength, but abruptly, and the critical value of the coupling strength depends on the frequency mismatch between the oscillators.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"992 - 1008"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145646106","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}

{"title":"On the Problem of Stability of Viscous Shocks","authors":"Sergey V. Bolotin,&nbsp;Dmitry V. Treschev","doi":"10.1134/S1560354725060024","DOIUrl":"10.1134/S1560354725060024","url":null,"abstract":"<div><p>We consider the problem of spectral stability of traveling wave solutions <span>(u=gamma(x-Wt))</span>\u0000for a system of viscous conservation laws <span>(partial_{t}u+partial_{x}F(u)=partial^{2}_{x}u)</span>.\u0000Such solutions correspond to heteroclinic trajectories <span>(gamma)</span> of a system of ODEs.\u0000In general conditions of stability can be obtained only numerically.\u0000We propose a model class of piecewise linear (discontinuous) vector fields <span>(F)</span> for which the\u0000stability\u0000problem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and construct\u0000several examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"908 - 930"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145646265","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}

{"title":"Normal Form of the Equations of Perturbed Motion near Triangular Libration Points\u0000at Third-Order Resonances","authors":"Anatoly P. Markeev","doi":"10.1134/S1560354725050053","DOIUrl":"10.1134/S1560354725050053","url":null,"abstract":"<div><p>A treatment is given of the spatial restricted elliptic problem of three\u0000bodies interacting under Newtonian gravity. The problem depends on two parameters:\u0000the ratio between the masses of the main attracting bodies and the eccentricity of their\u0000elliptic orbits. The eccentricity is assumed to be small. Nonlinear equations of motion of\u0000the test mass near a triangular libration point are analyzed. It is assumed that the\u0000parameters of the problem lie on the curves of third-order resonances corresponding to\u0000the planar restricted problem.\u0000In addition to these resonances (their number is equal to five), the spatial problem\u0000has a resonance that takes place at any parameter values since the\u0000the frequency of small linear oscillations of the test mass along the axis perpendicular to\u0000the plane of the orbit of the main bodies is equal to the frequency of Keplerian motion of\u0000these bodies.\u0000In this paper, the normal form of the Hamiltonian function of perturbed motion\u0000through fourth-degree terms relative to deviations from the libration point is obtained.\u0000Explicit expressions for the coefficients of normal form up to and including the second\u0000degree of eccentricity are found.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"837 - 846"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256069","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}

{"title":"On Pendulum-Type Motions and Permanent Rotations in an Approximate Problem of the Dynamics of a Rigid Body with a Vibrating Suspension","authors":"Mikhail V. Belichenko,&nbsp;Olga V. Kholostova","doi":"10.1134/S1560354725050065","DOIUrl":"10.1134/S1560354725050065","url":null,"abstract":"<div><p>The motion of a heavy rigid body with suspension point performing high-frequency periodic vibrations of small amplitude is considered. The study is carried out within the framework of an approximate autonomous system written in the form of the modified Euler – Poisson equations, to the right-hand sides of which the components of the vibration moment are added. The question of the existence of two particular motions of the body is resolved, they are permanent rotations and pendulum-type motions. It is shown that permanent rotations of the body can occur in the case of vibration symmetry relative to a vertically located axis. The search for pendulum-type motions is restricted to the case when the axis of these motions is one of the principal inertia axes of the body, as in the case of a heavy rigid body with a fixed point. Two basic variants of vibrations are considered, when the suspension point vibrates along a straight line and along an ellipse. To the latter variant any planar vibrations and a wide class of spatial vibrations of the suspension point are reduced.\u0000It is shown that for both basic cases of vibrations, pendulum-type motions are of two types. The motions of the first type are similar to the Mlodzeevsky’s pendulum-type motions of a heavy rigid body with a fixed point. For them, the body’s mass center lies in the principal plane of inertia, and the axis of the pendulum-type motions is perpendicular to this plane. Pendulum-type motions of the second type occur around the principal axis of inertia containing the body’s center of mass. Such motions are absent in the gravitational problem, they are caused by the presence of vibrations. To search for the pendulum-type motions, an approach is proposed that combines the\u0000results of the problem of gravitation (without vibration) and that of vibration\u0000(ignoring gravity). As an illustration, a number of examples of the interaction of the gravitational field and the vibration field corresponding to both basic variants of vibrations of the body’s suspension point are considered.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"847 - 865"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256068","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}

{"title":"When Knowledge of a Single Integral of Motion is Sufficient for Integration of Newton Equations (ddot{boldsymbol{q}}=boldsymbol{M}(boldsymbol{q}))","authors":"Stefan Rauch-Wojciechowski,&nbsp;Maria Przybylska","doi":"10.1134/S1560354725050077","DOIUrl":"10.1134/S1560354725050077","url":null,"abstract":"<div><p>For an autonomous dynamical system of <span>(n)</span> differential equations each integral of motion allows for reduction of the order of equations by 1 and knowledge of <span>((n-1))</span> integrals is necessary for the system to be integrated by quadratures. The amenability of Hamiltonian systems to being integrated by quadratures is characterised by the Liouville theorem where in <span>(2n)</span>-dimensional phase space only <span>(n)</span> integrals are sufficient as equations are generated by 1 function — the Hamiltonian.</p><p>There are, however, large families of Newton-type differential equations for which knowledge of 2 or 1 integral is sufficient for recovering separability and integration by quadratures. The purpose of this paper is to discuss a tradeoff between the number of integrals and the special structure of autonomous, velocity-independent 2nd order Newton equations <span>(ddot{boldsymbol{q}}=boldsymbol{M}(boldsymbol{q}))</span>, <span>(boldsymbol{q}inmathbb{R}^{n})</span>, which allows for integration by quadratures.</p><p>In particular, we review little-known results on quasipotential and triangular Newton equations to explain how it is possible that 2 or 1 integral is sufficient. The theory of these Newton equations provides new types of separation webs consisting of quadratic (but not orthogonal) surfaces.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"866 - 885"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256269","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}

{"title":"Contact Magnetic Geodesic and Sub-Riemannian Flows on (V_{n,2}) and Integrable Cases of a Heavy Rigid Body with a Gyrostat","authors":"Božidar Jovanović","doi":"10.1134/S156035472505003X","DOIUrl":"10.1134/S156035472505003X","url":null,"abstract":"<div><p>We prove the integrability of magnetic geodesic flows of <span>(SO(n))</span>-invariant Riemannian metrics on the rank two Stefel variety <span>(V_{n,2})</span> with respect to the magnetic field <span>(eta dalpha)</span>, where <span>(alpha)</span> is the standard contact form on <span>(V_{n,2})</span> and <span>(eta)</span> is a real parameter.\u0000Also, we prove the integrability of magnetic sub-Riemannian geodesic flows for <span>(SO(n))</span>-invariant sub-Riemannian structures on <span>(V_{n,2})</span>. All statements in the limit <span>(eta=0)</span> imply the integrability of the problems without the influence of the magnetic field. We also consider integrable pendulum-type natural mechanical systems with the kinetic energy defined by <span>(SO(n)times SO(2))</span>-invariant Riemannian metrics. For <span>(n=3)</span>, using the isomorphism <span>(V_{3,2}cong SO(3))</span>, the obtained integrable magnetic models reduce to\u0000integrable cases of the motion of a heavy rigid body with a gyrostat around a fixed point:\u0000the Zhukovskiy – Volterra gyrostat, the Lagrange top with a gyrostat, and the Kowalevski\u0000top with a gyrostat. As a by-product we obtain the Lax presentations for the Lagrange\u0000gyrostat and the Kowalevski gyrostat in the fixed reference frame (dual Lax representations).</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"799 - 818"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256267","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}

{"title":"On the 175th Anniversary of S. V. Kovalevskaya","authors":"","doi":"10.1134/S1560354725050016","DOIUrl":"10.1134/S1560354725050016","url":null,"abstract":"","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"765 - 766"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256270","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}