Peter De Maesschalck, Freddy Dumortier, Robert Roussarie
{"title":"Side-Comparison for Transition Maps in Multi-Layer Canard Problems","authors":"Peter De Maesschalck, Freddy Dumortier, 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}
{"title":"Three-Body Relative Equilibria on (mathbb{S}^{2})","authors":"Toshiaki Fujiwara, 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<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}
{"title":"On Phase at a Resonance in Slow-Fast Hamiltonian Systems","authors":"Yuyang Gao, Anatoly Neishtadt, 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}
{"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}
{"title":"Polynomial Entropy and Polynomial Torsion for Fibered Systems","authors":"Flavien Grycan-Gérard, 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}
{"title":"Compactification of the Energy Surfaces for (n) Bodies","authors":"Andreas Knauf, 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}
{"title":"A Simple Proof of Gevrey Estimates for Expansions of Quasi-Periodic Orbits: Dissipative Models and Lower-Dimensional Tori","authors":"Adrián P. Bustamante, 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}
Lluís Alsedà, David Juher, Jérôme Los, Francesc Mañosas
{"title":"On Families of Bowen – Series-Like Maps for Surface Groups","authors":"Lluís Alsedà, David Juher, Jérôme Los, Francesc Mañosas","doi":"10.1134/S1560354723040093","DOIUrl":"10.1134/S1560354723040093","url":null,"abstract":"<div><p>We review some recent results on a class of maps, called Bowen – Series-like maps, obtained from a class of group presentations for surface groups. These maps are piecewise homeomorphisms of the circle with finitely many discontinuities. The topological entropy of each map in the class and its relationship with the growth function of the group presentation is discussed, as well as the computation of these invariants.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"659 - 667"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50462505","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 Lambert Problem with Drag","authors":"Antonio J. Ureña","doi":"10.1134/S156035472304010X","DOIUrl":"10.1134/S156035472304010X","url":null,"abstract":"<div><p>The Lambert problem consists in connecting two given points in a given lapse of time under the gravitational influence of a fixed center. While this problem is very classical, we are concerned here with situations where friction forces act alongside the Newtonian attraction. Under some boundedness assumptions on the friction, there exists exactly one rectilinear solution if the two points lie on the same ray, and at least two solutions traveling in opposite directions otherwise.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"668 - 689"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50464491","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 Siegel – Bruno Linearization Theorem","authors":"Patrick Bernard","doi":"10.1134/S1560354723040147","DOIUrl":"10.1134/S1560354723040147","url":null,"abstract":"<div><p>The purpose of this paper is a pedagogical one. We provide a short and self-contained account of Siegel’s theorem, as improved by Bruno, which states that a holomorphic map of the complex plane can be locally linearized near a fixed point under certain conditions on the multiplier. The main proof is adapted from Bruno’s work.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"756 - 762"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50500795","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}
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