Volume 27, Number 1
Volume 27, Number 1, 2022
Alexey Borisov Memorial Volume
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 quasicoordinates 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.

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 singlevalued 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 singlevalued 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. 
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.

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 planeparallel 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 fixedposition 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.

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 $N$?

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.

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 saddlenode 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.

PérezRothen Y., Valeriano L., Vidal C.
On the Parametric Stability of the Isosceles Triangular Libration Points in the Planar Elliptical Charged Restricted Threebody Problem
Abstract
We consider the planar charged restricted elliptic threebody
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.
