Regular and Chaotic Dynamics最新文献

{"title":"Destruction of Invariant Curves and Singular Shilnikov Attractors in the Chialvo Map","authors":"Nataliya V. Stankevich,&nbsp;Alexander S. Gonchenko,&nbsp;Elena S. Popova,&nbsp;Elmira R. Bagautdinova","doi":"10.1134/S1560354725560059","DOIUrl":"10.1134/S1560354725560059","url":null,"abstract":"<div><p>We study bifurcation scenarios leading to the so-called Shilnikov singular attractors\u0000as a result of breakdown of closed invariant curves in the Chialvo map, which is a two-dimensional endomorphism demonstrating neuron-like dynamics. We show that two different routes, soft and hard, of the emergence of such closed invariant curves can be traced here: the soft one corresponds to the birth of an invariant\u0000curve as a result of a supercritical Neimark – Sacker bifurcation, and the hard one relates\u0000to the immediately occurring big invariant curve after disappearance of a stable fixed point\u0000at the saddle-node bifurcation. We study both these mechanisms and trace subsequent scenarios\u0000of breakdown of the invariant curve and chaos development leading to the emergence of\u0000Shilnikov singular attractors. Additionally, we study geometrical peculiarities of\u0000these attractors, such as structures of rotating patterns of orbits inside the Shilnikov\u0000singular funnel and present a two-parametric analysis with Lyapunov exponents and minimal\u0000a distance between the chaotic attractor and the unstable focus.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 Editors:","pages":"778 - 793"},"PeriodicalIF":0.8,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149214555","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 Mixed Dynamics in the Duffing Equation with Periodic Perturbations","authors":"Sergey V. Gonchenko,&nbsp;Kirill E. Morozov","doi":"10.1134/S1560354725560035","DOIUrl":"10.1134/S1560354725560035","url":null,"abstract":"<div><p>We study chaotic dynamics and bifurcations in reversible and time-periodic perturbations of the classical Duffing equation in the case when this equation has a homoclinic figure-8 to the saddle zero equilibrium.\u0000We consider the perturbation to be nonconservative and show that the phenomenon of mixed dynamics occurs in the corresponding Poincaré map <span>(T)</span> over the period.\u0000However, for small perturbations, the dynamics is predominantly dissipative: here, almost all orbits of <span>(T)</span> from the interior of the homoclinic-8 (perhaps, except for orbits from small resonant zones) tend to a stable fixed point (sink) at forward iterations and to an unstable fixed point (source) at backward iterations.\u0000We show that, when the amplitude of perturbation increases, mixed dynamics, as the phenomenon of the intersection of the attractor with the repeller, becomes quite noticeable and even prevalent. In the paper, we propose two different bifurcation scenarios that lead to such mixed dynamics. In the first scenario, it emerges as a result of the phenomenon known as the attractor-repeller collision. In the second scenario, mixed dynamics arises immediately after a symmetric pair of simple saddle-node bifurcations that, due to reversibility, occur simultaneously with the sink and the source.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 Editors:","pages":"53 - 68"},"PeriodicalIF":0.8,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147342876","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 Stationary (N)-Symmetric States of Three Point Vortices in a Two-Layer Rotating Fluid","authors":"Mikhail A. Sokolovskiy","doi":"10.1134/S1560354725550040","DOIUrl":"10.1134/S1560354725550040","url":null,"abstract":"<div><p>Within the framework of the two-layer quasi-geostrophic model on a rotating plane, the motions for a special case of three point vortices with intensities <span>(big{(}kappa_{1}^{1},kappa_{2}^{1},kappa_{2}^{2}big{)}=(4,1,-2))</span> are considered (here, the subscript denotes the layer number: 1 is the upper layer, 2 is the lower layer, and the superscript denotes the vortex number in the layer). Thus, it is assumed that one cyclonic vortex is located in the upper layer, and two vortices, cyclonic and anticyclonic, are located in the lower layer. It is shown that in the general case each vortex performs periodic motions in such a way that every half-period the vortex structure takes\u0000a collinear state. In this case, over time, the trajectory of each vortex completely fills\u0000a certain ring region around the vorticity center. However, among the continuum of\u0000these quasi-ordered trajectories, one can always find a family of closed periodic solutions,\u0000both purely circular (preserving the collinear structure) and more complex, so-called\u0000<span>(N)</span>-modal (<span>(N)</span>-symmetric) stationary solutions. In this paper, these solutions are constructed and their main properties are described.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 Editors:","pages":"116 - 131"},"PeriodicalIF":0.8,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147341567","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":"There are no Pseudo-Anosov Homeomorphisms of a Nonorientable Surface of Genus 3","authors":"Anatoly A. Medvedev","doi":"10.1134/S1560354725550039","DOIUrl":"10.1134/S1560354725550039","url":null,"abstract":"<div><p>In this paper, we prove the theorem given in the title and a similar theorem for generalized pseudo-Anosov homeomorphisms of the Klein bottle with only one “needle” singularity. We reformulate both statements in terms of the invariant foliation singularity type and orientability of foliations. The method we use to prove the theorems implies using the construction of band surface. Every band surface is represented with its combinatorial description, called the configuration. Applying a series of Rauzy transformations to all possible configurations in the cases considered, we show that the necessary conditions imposed on the invariant foliations are violated whence the results follow.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 3","pages":"407 - 425"},"PeriodicalIF":0.8,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148061150","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":"Synchronous Patterns in an Ensemble of Two Excitatory Coupled Nonidentical Excitable (phi)-Neurons","authors":"Alexander G. Korotkov,&nbsp;Tatiana A. Levanova,&nbsp;Grigory V. Osipov","doi":"10.1134/S1560354725550015","DOIUrl":"10.1134/S1560354725550015","url":null,"abstract":"<div><p>The purpose of this study is to investigate a simple model for the half-center oscillator (HCO), which consists of two nonidentical phase oscillators (<span>(phi)</span>-neurons) with mutual chemical excitatory couplings. In the absence of couplings, each element is in an excitable state. Using one- and two-parameter bifurcation analysis, we determine regions of stability of various synchronous temporal patterns typical for HCO and describe in detail bifurcation transitions between them. The proposed simple HCO model allows one to reproduce the main effects observed in biologically plausible HCO models, and can be easily implemented in locomotor units in neuromorphic robotics.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 3","pages":"453 - 467"},"PeriodicalIF":0.8,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148061149","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":"Liouvillian Solutions in the Problem of Motion of a Heavy Gyrostat with a Fixed Point under the Action of Gyroscopic Forces in the Hess Case","authors":"Alexander S. Kuleshov,&nbsp;Anton D. Skripkin","doi":"10.1134/S1560354725550027","DOIUrl":"10.1134/S1560354725550027","url":null,"abstract":"<div><p>The problem of motion of a heavy gyrostat with a fixed point under the action of gyroscopic forces, corresponding to the classical Hess case in the problem of motion of a heavy rigid body with a fixed point, is considered. We derive that the problem of motion of a gyrostat is reduced to solving the second-order linear differential equation with rational coefficients. Using the Kovacic algorithm, we obtain the conditions under which the general solution of the corresponding second-order linear differential equation is expressed in terms of Liouvillian functions and, therefore, it can be presented in explicit form. We prove that under the obtained conditions the equations of motion can be integrated by quadratures.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 Editors:","pages":"161 - 172"},"PeriodicalIF":0.8,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147341566","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 Rational Integrals of Geodesic Flows on 2-Surfaces","authors":"Sergei V. Agapov","doi":"10.1134/S1560354725540019","DOIUrl":"10.1134/S1560354725540019","url":null,"abstract":"<div><p>In this paper we study Riemannian metrics on 2-surfaces with integrable geodesic flows by\u0000means of an additional rational-in-momenta first integral. This problem is reduced to a quasi-linear system of PDEs. We construct solutions to this system via the classical hodograph method. These solutions give rise to local examples of metrics and rational integrals. Some of the constructed metrics have a very simple form. A family of implicit integrable examples parameterized by two arbitrary functions of one variable is also provided.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 Editors:","pages":"150 - 160"},"PeriodicalIF":0.8,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147340623","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":"Chaotic Dynamics and Stability Analysis of a Roller Bicycle","authors":"Ivan A. Bizyaev,&nbsp;Anna S. Berdnikova","doi":"10.1134/S1560354725540032","DOIUrl":"10.1134/S1560354725540032","url":null,"abstract":"<div><p>We analyze the stability of the straight-line motion of the bicycle depending on\u0000the mass-geometric parameters of the bicycle and its translational velocity.\u0000We construct a region in phase space which corresponds to initial conditions under which\u0000the bicycle tends asymptotically to straight-line motion. To investigate the\u0000bifurcations of the periodic solutions of the system, we construct a chart of\u0000dynamical regimes on the plane of two parameters and a three-dimensional Poincaré map.\u0000We analyze the possibility of acceleration or deceleration of the bicycle\u0000when the angular velocity of the rotor periodically changes\u0000in time.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 Editors:","pages":"28 - 52"},"PeriodicalIF":0.8,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147340184","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 Analytic Properties of the Perturbing Function in the PCR3Body Problem","authors":"Corrado Falcolini,&nbsp;Davide Zaccaria","doi":"10.1134/S1560354725540044","DOIUrl":"10.1134/S1560354725540044","url":null,"abstract":"<div><p>We provide a new expansion of the Fourier coefficient of the perturbing function of the\u0000PCR3Body problem in terms of Hansen coefficients. This gives us a precise asymptotic formula\u0000for the coefficient in the region of application of KAM theory (i. e., small value of eccentricity and semimajor\u0000axis. See, e. g., [17]). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes <span>((m,k)inmathbb{Z}^{2})</span> and the presence of common zeros as functions of actions between coefficients relative to modes <span>((m,k))</span>,<span>((2m,2k))</span> and <span>((m,k))</span>,<span>((2m,2k))</span>,<span>((3m,3k))</span>.\u0000Thanks to the previous expansion, this numerical analysis is done up to order <span>(60)</span> in the power\u0000of eccentricity and semimajor axis. This is the\u0000first step for a possible application of [4, 9] to the PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so-called “non-torus” set from <span>(O(1-sqrt{varepsilon}))</span> (implied by standard KAM theory) to <span>(O(1-varepsilon|logvarepsilon|^{c}))</span> for some <span>(c&gt;0)</span>.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 Editors:","pages":"92 - 115"},"PeriodicalIF":0.8,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147340626","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":"Comments on a Paper about Rubber Rolling by A. V. Borisov, I. S. Mamaev and I. A. Bizyaev (with an Appendix by Luis C. García-Naranjo)","authors":"Jair Koiller","doi":"10.1134/S1560354725540056","DOIUrl":"10.1134/S1560354725540056","url":null,"abstract":"<div><p>Rubber rolling (meaning no-slip and no-twist constraints) of a convex body on the plane under the influence of gravity is a <span>(SE(2))</span> Chaplygin system that reduces to the cotangent bundle of the unit sphere of Poisson vectors.\u0000I comment here upon an observation by A. V. Borisov and I. S. Mamaev [1, 2008], also found in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev [2, 2013] that <i>surfaces of revolution</i> are special: <i>the additional integral of motion is elementary, while for marble rolling it requires special functions</i>. I use the term “Nose function” to refer to their expression <span>(N(theta)=big{(}I_{1}cos^{2}theta+I_{3}sin^{2}theta+mz_{C}^{2}(theta)big{)}^{1/2})</span> where <span>(theta)</span> is the nutation and <span>(z_{C}(theta))</span> is the center of mass height. <span>(N(theta))</span> appears somewhat miraculously in the process of the almost symplectic reduction. I work in a space frame using the Euler angles <span>(phi text{(yaw)}, psi text{ roll and } theta)</span>. The reduction to 1 DoF is done in two stages: first, reduction by the group <span>(SE(2)={(x,y,phi)})</span> to <span>(T^{*}S^{2})</span> with almost symplectic 2-form <span>(Omega_{NH}=dp_{theta}wedge dtheta+dp_{psi}wedge dpsi+Jcdot K)</span>. The semibasic term is <span>(Jcdot K=-p_{psi}(dlogbig{(}N(theta)big{)}wedge dpsi)</span>. It follows that <span>(Omega_{NH})</span> is conformally symplectic in the sense that <span>(dleft(frac{1}{N}Omega_{NH}right)=0.)</span>\u0000The conserved quantity due to the <span>(S^{1})</span> symmetry about the body axis is <span>(ell=N(theta)sin^{2}thetadot{psi})</span>, yielding the desired reduction to <span>((theta,p_{theta}))</span>. Further simplification results by taking the new time <span>(dt=sqrt{B(theta)}dtau, text{ with } B=I_{1}+m|CP|^{2})</span> where <span>(P=(x,y))</span> is the point of contact.\u0000One gets finally <span>(H=frac{1}{2}tilde{p}^{2}_{theta}+V(theta),V(theta)=ell^{2}/2sin^{2}theta+mgz_{C}(theta) text{ with } tilde{p}_{theta}=p_{theta}/sqrt{B})</span> and usual symplectic form <span>(dtilde{p}_{theta}wedge dtheta)</span>. The moments of inertia <span>(I_{1},I_{3})</span> reappear in the reconstruction.\u0000As an example, very basic observations are presented for the torus.\u0000A detailed study was\u0000just finished by A. Kilin and E. Pivovarova in [3].</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 Editors:","pages":"1 - 27"},"PeriodicalIF":0.8,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147340629","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}