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Volume 27, Number 1

Volume 27, Number 1, 2022
Alexey Borisov Memorial Volume

ALEXEY BORISOV MEMORIAL VOLUME
Abstract
Citation: ALEXEY BORISOV MEMORIAL VOLUME, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 1-1
DOI:10.1134/S1560354722010014
Borisov A. V.,  Ivanov A. P.
A Top on a Vibrating Base: New Integrable Problem of Nonholonomic Mechanics
Abstract
A spherical rigid body rolling without sliding on a horizontal support is considered. The body is axially symmetric but unbalanced (tippe top). The support performs highfrequency oscillations with small amplitude. To implement the standard averaging procedure, we present equations of motion in quasi-coordinates in Hamiltonian form with additional terms of nonholonomicity [16] and introduce a new fast time variable. The averaged system is similar to the initial one with an additional term, known as vibrational potential [8, 9, 18]. This term depends on the single variable — the nutation angle $\theta$, and according to the work of Chaplygin [5], the averaged system is integrable. Some examples exhibit the influence of vibrations on the dynamics.
Keywords: nonholonomic mechanics, integrable system, oscillating support, tip-top
Citation: Borisov A. V.,  Ivanov A. P., A Top on a Vibrating Base: New Integrable Problem of Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 2-10
DOI:10.1134/S1560354722010026
Kozlov V. V.
On the Integrability of Circulatory Systems
Abstract
This paper discusses conditions for the existence of polynomial (in velocities)
first integrals of the equations of motion of mechanical systems in a nonpotential force field
(circulatory systems). These integrals are assumed to be single-valued smooth functions on
the phase space of the system (on the space of the tangent bundle of a smooth configuration
manifold). It is shown that, if the genus of the closed configuration manifold of such a system
with two degrees of freedom is greater than unity, then the equations of motion admit no
nonconstant single-valued polynomial integrals. Examples are given of circulatory systems with
configuration space in the form of a sphere and a torus which have nontrivial polynomial laws
of conservation. Some unsolved problems involved in these phenomena are discussed.
Keywords: circulatory system, polynomial integral, genus of surface
Citation: Kozlov V. V., On the Integrability of Circulatory Systems, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 11-17
DOI:10.1134/S1560354722010038
Tabachnikov S.
Remarks on Rigidity Properties of Conics
Abstract
Inspired by the recent results toward Birkhoff conjecture (a rigidity property of billiards in ellipses), we discuss two rigidity properties of conics. The first one concerns symmetries of an analog of polar duality associated with an oval, and the second concerns properties of the circle map associated with an oval and two pencils of lines.
Keywords: conics, polar duality, rigidity, circle maps, chess billiards
Citation: Tabachnikov S., Remarks on Rigidity Properties of Conics, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 18-23
DOI:10.1134/S156035472201004X
Artemova E. M.,  Vetchanin E. V.
The Motion of an Unbalanced Circular Disk in the Field of a Point Source
Abstract
Describing the phenomena of the surrounding world is an interesting task that has long attracted the attention of scientists. However, even in seemingly simple phenomena, complex dynamics can be revealed. In particular, leaves on the surface of various bodies of water exhibit complex behavior. This paper addresses an idealized description of the mentioned phenomenon. Namely, the problem of the plane-parallel motion of an unbalanced circular disk moving in a stream of simple structure created by a point source (sink) is considered. Note that using point sources, it is possible to approximately simulate the work of skimmers used for cleaning swimming pools. Equations of coupled motion of the unbalanced circular disk and the point source are derived. It is shown that in the case of a fixed-position source of constant intensity the equations of motion of the disk are Hamiltonian. In addition, in the case of a balanced circular disk the equations of motion are integrable. A bifurcation analysis of the integrable case is carried out. Using a scattering map, it is shown that the equations of motion of the unbalanced disk are nonintegrable. The nonintegrability found here can explain the complex motion of leaves in surface streams of bodies of water.
Keywords: ideal fluid, motion in the presence of a source, nonintegrability, scattering map, chaotic scattering
Citation: Artemova E. M.,  Vetchanin E. V., The Motion of an Unbalanced Circular Disk in the Field of a Point Source, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 24-42
DOI:10.1134/S1560354722010051
Chenciner A.,  Sauzin D.,  Sun S.,  Wei Q.
Elliptic Fixed Points with an Invariant Foliation: Some Facts and More Questions
Abstract
We address the following question: let $F:(\mathbb {R}^2,0)\to(\mathbb {R}^2,0)$ be an analytic local diffeomorphism defined in the neighborhood of the nonresonant elliptic fixed point 0 and let $\Phi$ be a formal conjugacy to a normal form $N$. Supposing $F$ leaves invariant the foliation by circles centered at $0$, what is the analytic nature of $\Phi$ and $N$?
Keywords: normal form, Arnold family, weakly attracting fixed point
Citation: Chenciner A.,  Sauzin D.,  Sun S.,  Wei Q., Elliptic Fixed Points with an Invariant Foliation: Some Facts and More Questions, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 43-64
DOI:10.1134/S1560354722010063
Polekhin I. Y.
The Spherical Kapitza – Whitney Pendulum
Abstract
In this paper we study the global dynamics of the inverted spherical pendulum with a vertically rapidly vibrating suspension point in the presence of an external horizontal periodic force field. We do not assume that this force field is weak or rapidly oscillating. Provided that the period of the vertical motion and the period of the horizontal force are commensurate, we prove that there always exists a nonfalling periodic solution, i. e., there exists an initial condition such that, along the corresponding solution, the rod of the pendulum always remains above the horizontal plane passing through the pivot point. We also show numerically that there exists an asymptotically stable nonfalling solution for a wide range of parameters of the system.
Keywords: forced oscillations, Kapitza pendulum, Whitney pendulum, stabilization, vibrations
Citation: Polekhin I. Y., The Spherical Kapitza – Whitney Pendulum, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 65-76
DOI:10.1134/S1560354722010075
Medvedev T. V.,  Nozdrinova E. V.,  Pochinka O. V.
Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms
Abstract
In 1976 S.Newhouse, J.Palis and F.Takens introduced a stable arc joining two structurally stable systems on a manifold. Later in 1983 they proved that all points of a regular stable arc are structurally stable diffeomorphisms except for a finite number of bifurcation diffeomorphisms which have no cycles, no heteroclinic tangencies and which have a unique nonhyperbolic periodic orbit, this orbit being the orbit of a noncritical saddle-node or a flip which unfolds generically on the arc. There are examples of Morse – Smale diffeomorphisms on manifolds of any dimension which cannot be joined by a stable arc. There naturally arises the problem of finding an invariant defining the equivalence classes of Morse – Smale diffeomorphisms with respect to connectedness by a stable arc. In the present review we present the classification results for Morse – Smale diffeomorphisms with respect to stable isotopic connectedness and obstructions to existence of stable arcs including the authors’ recent results.
Keywords: stable arc, Morse – Smale diffeomorphism
Citation: Medvedev T. V.,  Nozdrinova E. V.,  Pochinka O. V., Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 77-97
DOI:10.1134/S1560354722010087
Pérez-Rothen Y.,  Valeriano L.,  Vidal C.
On the Parametric Stability of the Isosceles Triangular Libration Points in the Planar Elliptical Charged Restricted Three-body Problem
Abstract
We consider the planar charged restricted elliptic three-body problem (CHRETBP). In this work we consider the parametric stability of the isosceles triangle equilibrium solution denoted by $L_4^{iso}$. We construct the boundary surfaces of the stability/instability regions in the space of the parameters $\mu$, $\beta$ and $e$, which are parameters of the mass, charges associated to the primaries and the eccentricity of the elliptic orbit, respectively.
Keywords: charged problem, restricted three-body problem, parametric stability
Citation: Pérez-Rothen Y.,  Valeriano L.,  Vidal C., On the Parametric Stability of the Isosceles Triangular Libration Points in the Planar Elliptical Charged Restricted Three-body Problem, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 98-121
DOI:10.1134/S1560354722010099

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