 EDITORINCHIEF
 Honorary Editor
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Gasanov M. V., Gulkanov A. G., Modestov K. A.
Abstract
In this paper, we consider a mathematical model of the dynamics of the behavior of a spherically
symmetric Rayleigh – Plesset bubble in the van der Waals gas model. The analysis of the
model takes into account various isoprocesses without the presence of condensation and a model
that takes into account condensation in an isothermal process. In each case, various characteristics
are searched for, such as oscillation frequency (linear/small oscillations), damping factor,
relaxation time, decrement, and logarithmic decrement. Solutions are found in quadratures for
various parameters of the equation. The theoretical results obtained are compared with the
results of the numerical solution of the Cauchy problem for various isoprocesses.

Fakhretdinov M. I., Samsonov K. Y., Dmitriev S. V., Ekomasov E. G.
Abstract
The $\varphi^4$ theory is widely used in many areas of physics, from cosmology and elementary particle physics to biophysics and condensed matter theory. However, in the $\varphi^4$ model, there are no spatially localized solutions in the form of breathers. Topological defects, or kinks, in this theory describe stable, solitary wave excitations. In practice, these excitations, as they propagate, necessarily interact with impurities or imperfections in the onsite potential. In this work, with the help of numerical calculations using the method of lines, the interaction of the kink in the $\varphi^4$ model with extended impurities is considered. The case of an attractive rectangular impurity is analyzed. It is found that after the kinkimpurity interaction, an internal mode with frequency $\sqrt{\frac32}$ is excited on the kink and it becomes a wobbling kink. It is shown that with the help of kinkimpurity interaction, an extended rectangular attracting impurity, as well as a point impurity, can be used as a generator for excitation of longlived highamplitude localized breather waves. The structure of the excited wobbling breather (or wobbler), which consists of a compact core and an extended tail, is described. It is shown that the wobbler tail has the form of a spatially unbounded quasisinusoidal function with a classical frequency $\sqrt{2}$. To determine the lifetime of the wobbler, the dependence of the amplitude of the impurity mode on time is found. For the case of small impurities, it turned out that it practically does not change for a long time. For the case of large impurities, the wobbler amplitude begins to noticeably decrease with time. The frequency of wobbler oscillations does not depend on the initial velocity of the kink. The dependence of the impurity mode oscillation amplitude on the initial kink velocity has minima and maxima. By changing the impurity parameters, one can also control the dynamic parameters of the wobbler. A linear approximation is considered that allows an analytical solution of the problem for localized breather waves, and the limits of its applicability for this model are found.

Ardentov A. A.
Abstract
We formulate a timeoptimal problem for a differential drive robot with bounded positive
velocities of the driving wheels. This problem is equivalent to a generalization of the classical
Markov – Dubins problem with an extended domain of control. We classify all extremal controls
via the Pontryagin maximum principle. Some optimality conditions are obtained; therefore, the
optimal synthesis is reduced to the enumeration of a finite number of possible solutions.

Adamov B. I.
Abstract
This article is devoted to a study of the geometry and kinematics of the Mecanum wheels,
also known as Ilon wheels or the Swedish wheels. The Mecanum wheels are one of the types of
omnidirectional wheels. This property is provided by peripheral rollers whose axes are deviated
from the wheel one by 45 degrees. A unified approach to studying the geometry and kinematics
of the Mecanum wheels on a plane and on the internal or external surface of a sphere is proposed.
Kinematic relations for velocities at the contact point of the wheel and the supporting surface, and
angular velocities of the roller relative to the supporting surface are derived. They are necessary
to describe the dynamics of the Mecanum systems taking into account forces and moments of
contact friction in the presence of slipping. From the continuous contact condition, relations
determining the geometry of the wheel rollers on a plane and on the internal or external surface
of a sphere are obtained. The geometric relations for the Mecanum wheel rollers could help to
adjust the existing shape of the Mecanum wheel rollers of spherical robots and ballbots to improve
the conditions of contact between the rollers and the spherical surface. An analytical study of
the roller geometry was carried out, and equations of their generatrices were derived. Under the
noslipping condition, expressions for rotational velocities of the wheel and the contacting roller
are obtained. They are necessary for analyzing the motion of systems within the framework
of nonholonomic models, solving problems of controlling Mecanum systems and improving its
accuracy. Using the example of a spherical robot with an internal threewheeled Mecanum
platform, the influence of the rollers on the robot movement was studied at the kinematic level.
It has been established that the accuracy of the robot movement is influenced not only by
the geometric parameters of the wheels and the number of rollers, but also by the relationship
between the components of the platform center velocity and its angular velocity. Results of the
numerical simulation of the motion of the spherical robot show a decrease in control accuracy in
the absence of feedback on the robot’s position due to effects associated with the finite number
of rollers, their geometry and switching. These effects lead not only to highfrequency vibrations,
but also to a “drift” of the robot trajectory relative to the reference trajectory. Further research
on this topic involves the use of the motion separation methods and the statistical methods for
kinematical and dynamical analysis of Mecanum systems.

Vetchanin E. V., Valieva A. R.
Abstract
The problem of describing the motion of a rigid body in a fluid is addressed by considering
a symmetric Joukowsky foil. Within the framework of the model of an ideal fluid, the force and
torque acting on an unsteady moving foil are calculated. The analytical results are compared
with those obtained based on the numerical solution of the Navier – Stokes equations. It is shown
that analytical expressions for the force and torque can be consistent with the results of numerical
simulations using scaling and a delayed arguments.

Kilin A. A., Artemova E. M.
Abstract
This paper addresses the problem of the motion of two point vortices of arbitrary strengths
in an ideal incompressible fluid on a finite flat cylinder. A procedure of reduction to the level set
of an additional first integral is presented. It is shown that, depending on the parameter values,
three types of bifurcation diagrams are possible in the system. A complete bifurcation analysis of
the system is carried out for each of them. Conditions for the orbital stability of generalizations
of von Kármán streets for the problem under study are obtained.

Gaydukov R. K.
Abstract
The problem of flow of a nonNewtonian viscous fluid with powerlaw rheological properties
along a semiinfinite plate with a small localized irregularity on the surface is considered for large
Reynolds numbers. The asymptotic solution with doubledeck structure of the boundary layer is
constructed. The numerical simulation of the flow in the region near the surface was performed
for different fluid indices. The results of investigations of the flow properties depending on the
fluid index are presented. Namely, the boundary layer separation is investigated for different
fluid indices, and the dynamics of vortex formation in this region is shown.

Bardin B. S.
Abstract
The problem of the orbital stability of periodic motions of a heavy rigid body with a fixed
point is investigated. The periodic motions are described by a particular solution obtained by
D. N. Bobylev and V. A. Steklov and lie on the zero level set of the area integral. The problem of
nonlinear orbital stability is studied. It is shown that the domain of possible parameter values
is separated into two regions: a region of orbital stability and a region of orbital instability. At
the boundary of these regions, the orbital instability of the periodic motions takes place.

Ivanov A. P.
Abstract
We consider the planar problem of the dynamics of a body moving along a horizontal with
two legs in contact with a rough horizontal plane. Possible types of movements of the body are
discussed depending on the acceleration of the support: relative equilibrium, sliding on two legs,
lifting off one leg without sliding on the other, lifting off one leg while sliding on the other. Based
on the results obtained, it is shown that, when sliding on two legs, the friction force is generally
anisotropic. This makes it possible to transport the body due to simple vibrations of the plane,
for example, harmonic vibrations.

Gonchenko S. V., Gordeeva O. V.
Abstract
We consider twodimensional diffeomorphisms with homoclinic orbits to nonhyperbolic fixed
points. We assume that the point has arbitrary finite order degeneracy and is either of saddlenode
or weak saddle type. We consider two cases when the homoclinic orbit is transversal and
when a quadratic homoclinic tangency takes place. In the first case we give a complete description
of orbits entirely lying in a small neighborhood of the homoclinic orbit. In the second case we
study main bifurcations in oneparameter families that split generally the homoclinic tangency
but retain the degeneracy type of the fixed point.

Osenkov E. M., Pochinka O. V.
Abstract
In this paper, we consider a class of Morse – Smale diffeomorphisms defined on a closed
3manifold (not necessarily orientable) under the assumption that all their saddle points have
the same dimension of the unstable manifolds. The simplest example of such diffeomorphisms is
the wellknown “sourcesink” or “north pole – south pole” diffeomorphism, whose nonwandering
set consists of exactly one source and one sink. As Reeb showed back in 1946, such systems can
only be realized on the sphere. We generalize his result, namely, we show that diffeomorphisms
from the considered class also can be defined only on the 3sphere.

Mikishanina E. A.
Abstract
This study investigates the rolling along the horizontal plane of two coupled rigid bodies:
a spherical shell and a dynamically asymmetric rigid body which rotates around the geometric
center of the shell. The inner body is in contact with the shell by means of omniwheels.
A complete system of equations of motion for an arbitrary number of omniwheels is constructed.
The possibility of controlling the motion of this mechanical system along a given trajectory by
controlling the angular velocities of omniwheels is investigated. The cases of two omniwheels
and three omniwheels are studied in detail. It is shown that two omniwheels are not enough to
control along any given curve. It is necessary to have three or more omniwheels. The quaternion
approach is used to study the dynamics of the system.
