 EDITORINCHIEF
 Honorary Editor
 Editorial board

 Passed away
Berestova S. A., Prosviryakov E. Y.
Abstract
An exact solution of the Oberbeck – Boussinesq equations for the description of the steadystate
Bénard – Rayleigh convection in an infinitely extensive horizontal layer is presented. This
exact solution describes the largescale motion of a vertical vortex flow outside the field of the
Coriolis force. The largescale fluid flow is considered in the approximation of a thin layer with
nondeformable (flat) boundaries. This assumption allows us to describe the largescale fluid
motion as shear motion. Two velocity vector components, called horizontal components, are taken
into account. Consequently, the third component of the velocity vector (the vertical velocity) is
zero. The shear flow of the vertical vortex flow is described by linear forms from the horizontal
coordinates for velocity, temperature and pressure fields. The topology of the steady flow of
a viscous incompressible fluid is defined by coefficients of linear forms which have a dependence
on the vertical (transverse) coordinate. The functions unknown in advance are exactly defined
from the system of ordinary differential equations of order fifteen. The coefficients of the forms
are polynomials. The spectral properties of the polynomials in the domain of definition of the
solution are investigated. The analysis of distribution of the zeroes of hydrodynamical fields has
allowed a definition of the stratification of the physical fields. The paper presents a detailed study
of the existence of steady reverse flows in the convective fluid flow of Bénard – Rayleigh – Couette
type.

Rozenblat G. M., Grishakin V. T.
Abstract
This paper deals with a formulation and a solution of problems of the dynamics of mechanical
systems for which solutions that do not take into account the unilateral nature of the constraints
imposed on the objects under study have been obtained before. The motive force in all the
cases considered is the gravity force applied to the center of mass of each body of the mechanical
system. Since unilateral constraints are imposed on all systems of bodies considered in the abovementioned
problems, their correct solution requires taking into account the unilateral action of
the constraint reaction forces applied to the bodies of the systems under study. A detailed
analysis of the motion of the systems after zeroing out the constraint reaction forces is carried
out. Results of numerical experiments are presented which are used to construct motion patterns
of the systems of bodies illustrating the motions of the abovementioned systems after they lose
contact with the supporting surfaces.

Ledkov A., Pikalov R.
Abstract
Tether retrieval is an important stage in many projects using space tether systems. It is
known that uniform retrieval is an unstable process that leads to the winding of the tether on
a satellite at the final stage of retraction. This is a serious obstacle to the practical application
of space tethers in the tasks of climbing payloads to a satellite and docking the spacecraft with
a tethered satellite after its capture. The paper investigates the plane motion of a space tether
system with a massless tether of variable length in an elliptical orbit. A new control law that
ensures the retrieval of the tether without increasing the amplitude of oscillations at the final
stage is proposed. The asymptotic stability of the space tether system’s controlled motion in an
elliptical orbit is proved. A numerical analysis of tether retrieval is carried out. The influence of
the eccentricity of the orbit on the retrieval process is investigated. The results of the work can
be useful in preparing missions of the active space debris removal and in performing operations
involving tether retrieval.

Shvarts K. G.
Abstract
In this paper a new exact solution of the Navier – Stokes equations in the Boussinesq approximation
describing advective flow in a horizontal liquid layer with free boundaries, where the
vertical velocity component is a constant value, is obtained. The temperature is linear along the
boundaries of the layer. Solutions of this kind are used to close threedimensional equations averaged
across the layer in the derivation of twodimensional models of nonisothermal largescale
flows in a thin layer of liquid or incompressible gas. The properties of advective flow at different
values of Reynolds number and Prandtl number are investigated.

Medvedev V. S., Zhuzhoma E. V.
Abstract
We prove that, given any $n\geqslant 3$ and $2\leqslant q\leqslant n1$, there is a closed $n$manifold $M^n$ admitting a chaotic lamination of codimension $q$ whose support has the topological dimension ${nq+1}$. For $n=3$ and $q=2$, such chaotic laminations can be represented as nontrivial 2dimensional basic sets of axiom A flows on 3manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repellerattractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with onedimensional basic sets obtained previously by V. Grines, R. Plykin and Yu. Zhirov [16, 28, 31].

Soga K.
Abstract
This paper provides a quite simple method of Tonelli’s calculus of variations with positive
definite and superlinear Lagrangians. The result complements the classical literature of calculus
of variations before Tonelli’s modern approach. Inspired by Euler’s spirit, the proposed method
employs finitedimensional approximation of the exact action functional, whose minimizer is easily
found as a solution of Euler’s discretization of the exact Euler – Lagrange equation. The
Euler – Cauchy polygonal line generated by the approximate minimizer converges to an exact
smooth minimizing curve. This framework yields an elementary proof of the existence and regularity
of minimizers within the family of smooth curves and hence, with a minor additional step,
within the family of Lipschitz curves, without using modern functional analysis on absolutely
continuous curves and lower semicontinuity of action functionals.

Gorr G. V.
Abstract
This paper is concerned with a special class of precessions of a rigid body having a fixed
point in a force field which is a superposition of three homogeneous force fields. It is assumed
that the velocity of proper rotation of the body is twice as large as its velocity of precession. The
conditions for the existence of the precessions under study are written in the form of a system of
algebraic equations for the parameters of the problem. Its solvability is proved for a dynamically
symmetric body. It is proved that, if the ellipsoid of inertia of the body is a sphere, then the
nutation angle is equal to $\arccos \frac{1}{3}$. The resulting solution of the equations of motion of the body
is represented as elliptic Jacobi functions.

Klekovkin A. V., Karavaev Y. L., Mamaev I. S.
Abstract
This paper presents the design of an aquatic robot actuated by one internal rotor. The robot
body has a cylindrical form with a base in the form of a symmetric airfoil with a sharp edge. For
this object, equations of motion are presented in the form of Kirchhoff equations for rigid body
motion in an ideal fluid, which are supplemented with viscous resistance terms. A prototype
of the aquatic robot with an internal rotor is developed. Using this prototype, experimental
investigations of motion in a fluid are carried out.

Moskvitin V. M., Semenova N. I.
Abstract
In recent years, more and more researchers in the field of artificial neural networks have
been interested in creating hardware implementations where neurons and the connection between
them are realized physically. Such networks solve the problem of scaling and increase the speed
of obtaining and processing information, but they can be affected by internal noise. In this paper we analyze an echo state neural network (ESN) in the presence of uncorrelated additive and multiplicative white Gaussian noise. Here we consider the case where artificial neurons have a linear activation function with different slope coefficients. We consider the influence of the input signal, memory and connection matrices on the accumulation of noise. We have found that the general view of variance and the signaltonoise ratio of the ESN output signal is similar to only one neuron. The noise is less accumulated in ESN with a diagonal reservoir connection matrix with a large “blurring” coefficient. This is especially true of uncorrelated multiplicative noise. 