 COEDITORSINCHIEF
 Honorary Editor
 Editorial board

Vol. 16, no. 2
Yehia H. M., Hussein A. M.
Abstract
The method introduced in [11] and [12] is extended to construct new families of severalparameter
integrable systems, which admit a complementary integral quartic in the velocities.
A list of 14 systems is obtained, of which 12 are new. Each of the new systems involves a number
of parameters ranging from 7 up to 16 parameters entering into its structures. A detailed
preliminary analysis of certain special cases of one of the new systems is performed, aimed at
obtaining some global results. We point out twelve combinations of conditions on the parameters
which characterize integrable dynamics on Riemannian manifolds as configuration spaces. Very
special 7 versions of the 12 cases are interpreted as new integrable motions with a quartic
integral in the Poincaré halfplane. A byproduct of the process of solution is the construction
of 12 Riemannian metrics whose geodesic flow is integrable with a quartic second integral.

Bukh A. V., Anishchenko V. S.
Abstract
The features of external and mutual synchronization of spiral wave structures including
chimera states in the interacting twodimensional lattices of nonlocally coupled Nekorkin maps
are investigated. The cases of diffusive and inertial couplings between the lattices are considered.
The lattices model a neuronal activity and represent twodimensional lattices consisting
of $N \times N$ elements with $N = 200$. It is shown that the effect of complete synchronization is
not achieved in the studied lattices, and only the regime of partial synchronization is realized
regardless of the case of coupling between the lattices. It is important to note that the conclusion
is applied not only to the regimes of spiral wave chimeras, but also to the regimes of regular
spiral waves.

Burov A. A., Nikonov V. I.
Abstract
As is well known, many small celestial bodies are of a rather complex shape. Therefore, the
study of the dynamics of a spacecraft in their vicinity, based on terms up to the second order of
smallness in the expansion of the potential of attraction, seems to be insufficient for an adequate
description of the observed dynamical effects related, for example, to positioning of the libration
points. In this paper, such effects are demonstrated for spacecraft dynamics in the vicinity of the asteroid (2063) Bacchus. The libration points are computed for various approximations of the gravitational potential. The results of this computation are compared with similar results obtained before for the socalled Sludsky – Werner – Scheeres potential. The dependence of the structure of the regions of possible motions on approximation of the gravitational potential is also studied. 
Mogilevich L., Ivanov S., Blinkov Y.
Abstract
This article investigates longitudinal deformation waves in physically nonlinear coaxial elastic
shells containing a viscous incompressible fluid between them. The rigid nonlinearity of the
shells is considered. The presence of a viscous incompressible fluid between the shells, as well
as the influence of the inertia of the fluid motion on the amplitude and velocity of the wave, are
taken into account. A numerical study of the model constructed in the course of this work is carried out by using a difference scheme for the equation similar to the Crank – Nicolson scheme for the heat equation. In the case of identical initial conditions in both shells, the deformation waves in them do not change either the amplitude or the velocity. In the case of setting different initial conditions in the coaxial shells, the amplitude of the solitary wave in the first shell decreases from the value specified at the initial instant of time, and in the second, the amplitude grows from zero until they equalize, that is, energy is transferred. The movement occurs in a negative direction. This means that the velocity of deformation wave is subsonic. 
Adamov B. I., Saypulaev G. R.
Abstract
The subject of this study is an omnidirectional mobile platform equipped with four
Mecanum wheels. The movement of the system on a horizontal plane is considered. The aim of
this research is to study the dynamics of the omnidirectional platform, taking into account the
design of Mecanum wheels: the shape of the rollers and their finite number. The equations of
motion of the onmidirectional mobile platform are derived taking into account the real design
of the Mecanum wheels and their slippage. A comparative analysis of the results of numerical
modeling for different models of contact friction forces is presented. It has been established
that switching of contact rollers and displacement of contact points lead to the occurrence of
highfrequency components of wheel rotation speeds, as well as an offset of their average values
(in comparison with the modeling results without taking into account the design features of the
chassis).

Essaouini H., Capodanno P.
Abstract
We study the small oscillations of a pendulum completely filled by a viscoelastic fluid,
restricting ourselves for the fluid to the simpler Oldroyd model. We establish the equations
of motion of the system. Writing them in a suitable form, we obtain an existence and unicity
theorem of the solution of the associated evolution problem by means of semigroup theory.
Afterwards, we show the existence and symmetry of the spectrum and prove the stability of the
system. We show the existence of two sets of positive real eigenvalues, of which the first has
infinity, and the second a point of the real axis, as points of accumulation. Finally, we specify
the location of the possible nonreal eigenvalues.

Kuzenov V. V., Ryzhkov S. V., Starostin A. V.
Abstract
This work is devoted to the theoretical calculation of the processes of compression and energy
release in the target by a combined action of a system of pulsed jets and intense laser radiation
using a magnetic inertial plasma confinement method. A mathematical model, a numerical
method, and a computational algorithm are developed to describe plasmaphysical processes
occurring in various types of hightemperature installations with high density. The results of
the calculation of the hybrid effect of intensive energy flows on a cylindrical target are presented.
The main gasdynamic and radiative parameters of the compressed target plasma are found.

Polekhin I. Y.
Abstract
In this paper we study the existence of forced oscillations in two Lagrange systems with
gyroscopic forces: a spherical pendulum in a magnetic field and a point on a rotating closed
convex surface. We show how it is possible to prove the existence of forced oscillations in these
systems provided the systems move in the presence of viscous friction.

Sachkov Y. L.
Abstract
The Engel group is the fourdimensional nilpotent Lie group of step 3, with 2 generators.
We consider a oneparameter family of leftinvariant rank 2 subFinsler problems on the Engel
group with the set of control parameters given by a square centered at the origin and rotated
by an arbitrary angle. We adopt the viewpoint of timeoptimal control theory. By Pontryagin’s
maximum principle, all subFinsler length minimizers belong to one of the following types:
abnormal, bangbang, singular, and mixed. Bangbang controls are piecewise controls with
values in the vertices of the set of control parameters. We describe the phase portrait for bangbang extremals. In previous work, it was shown that bangbang trajectories with low values of the energy integral are optimal for arbitrarily large times. For optimal bangbang trajectories with high values of the energy integral, a general upper bound on the number of switchings was obtained. In this paper we improve the bounds on the number of switchings on optimal bangbang trajectories via a secondorder necessary optimality condition due to A. Agrachev and R.Gamkrelidze. This optimality condition provides a quadratic form, whose signdefiniteness is related to optimality of bangbang trajectories. For each pattern of these trajectories, we compute the maximum number of switchings of optimal control. We show that optimal bangbang controls may have not more than 9 switchings. For particular patterns of bangbang controls, we obtain better bounds. In such a way we improve the bounds obtained in previous work. On the basis of the results of this work we can start to study the cut time along bangbang trajectories, i.e., the time when these trajectories lose their optimality. This question will be considered in subsequent work. 
Morozov A. D., Morozov K. E.
Abstract
We study nonconservative quasiperiodic $m$frequency $\it parametric$ perturbations of twodimensional
nonlinear Hamiltonian systems. Our objective is to specify the conditions for the
existence of new regimes in resonance zones, which may arise due to parametric terms in the
perturbation. These regimes correspond to $(m + 1)$frequency quasiperiodic solutions, which
are not generated from Kolmogorov tori of the unperturbed system. The conditions for the
existence of these solutions are found. The study is based on averaging theory and the analysis
of the corresponding averaged systems. We illustrate the results with an example of a Duffing
type equation.

Kugushev E. I., Popova T. V.
Abstract
We consider systems of ordinary differential equations whose righthand sides contain timeperiodic
functions with some frequencies. An averaged system is constructed by introduction of
additional variables and by stepbystep averaging over these variables. An upper estimate of
the deviation of the solution to the initial system from the solution to the averaged system is
given. Examples are given of mechanical systems in which vibrations with several frequencies
occur and for the analysis of which the statements obtained are applied.
