 COEDITORSINCHIEF
 Honorary Editor
 Editorial board

Call for papers
Call for Papers: Special Issue dedicated to the 150th birthday of Sergey A. Chaplygin 
Vol. 15, no. 4
Adabrah A. K., Dragović V., Radnović M.
Abstract
We derive necessary and sufficient conditions for periodic trajectories of billiards within an
ellipse in the Minkowski plane in terms of an underlying elliptic curve. Equivalent conditions
are derived in terms of polynomialfunctional equations as well. The corresponding polynomials
are related to the classical extremal polynomials. Similarities and differences with respect to
the previously studied Euclidean case are indicated.

Ardentov A. A.
Abstract
This investigation continues the study of the classical problem of stationary configurations
of an elastic rod on a plane. The length of the rod, the ends of the rod and the directions at the
ends are fixed. The problem was first studied by Leonard Euler in 1744 and the optimal synthesis
problem is still an open problem. Euler described a family of geodesics containing the solutions,
which are called Euler elasticae. It is known that sufficiently small pieces of Euler elasticae
are optimal, i.e., they have a minimum of the potential energy. In theory, the point where an
optimal curve loses its optimality is called a cut point. Usually several optimal curves arrive at
such points, so the points have multiplicity more than 1 and are called Maxwell points. The aim
of this work is to describe numerically Maxwell points where two nonsymmetric elasticae come
with the same length and energy value.

Bardin B. S., Chekina E.
Abstract
The motion of a rigid body satellite about its center of mass is considered. The problem
of the orbital stability of planar pendulumlike rotations of the satellite is investigated. It is
assumed that the satellite moves in a circular orbit and its geometry of mass corresponds to
a plate. In unperturbed motion the minor axis of the inertia ellipsoid lies in the orbital plane. A nonlinear analysis of the orbital stability for previously unexplored values of parameters corresponding to the boundaries of the stability regions is carried out. The study is based on the normal form technique. In the special case of fast rotations a normalization procedure is performed analytically. In the general case the coefficients of normal form are calculated numerically. It is shown that in the case under consideration the planar rotations of the satellite are mainly unstable, and only on one of the boundary curves there is a segment where the formal orbital stability takes place. 
Bountis A., Kominis Y., Shena J., Kovanis V.
Abstract
Coupled semiconductor lasers are systems possessing complex dynamics, which makes them
interesting for many applications in photonics. In this paper, we first review our results on the
existence and stability of asymmetric phaselocked states of a single dimer consisting of two
coupled semiconductor lasers. We show that stable phaselocked states of arbitrary asymmetry
exist, whose field amplitude ratio and phase difference can be dynamically controlled by appropriate
electronic current injection. Moreover, we obtain stable limit cycles with asymmetric
characteristics, emerging through Hopf bifurcations from these phaselocked states. Also, we
emphasize the importance of exceptional points, and we show that asymmetry enables their existence
in extended regions of parameter space. The dynamics of asymmetric dimers under small
signal modulation of the pumping current is also investigated and the occurrence of antiresonances
and sharp resonances with very high frequencies is demonstrated. Finally, we describe
our recent findings on optically coupled arrays of coupled dimers and explore their fascinating
nonlinear dynamics. In particular, we couple in an appropriate way a large number of dimers
and show that, depending on their degree of asymmetry, they exhibit organized high amplitude
oscillations, or oscillate very close to phaselocked states, suggesting that such photonic networks
may prove useful in a variety of beam forming and beam shaping applications.

Gajić B., Jovanović B.
Abstract
We consider the nonholonomic problem of rolling without
slipping and twisting of a balanced ball over a fixed sphere in $\mathbb{R}^n$.
By relating the system to a modified LR system, we prove that the problem always has an invariant measure. Moreover,
this is a~$SO(n)$Chaplygin system that reduces to the cotangent bundle $T^*S^{n1}$.
We present two integrable cases. The first one is obtained for a special inertia operator that allows the Chaplygin Hamiltonization of the reduced system.
In the second case, we consider the rigid body inertia operator
$\mathbb I\omega=I\omega+\omega I$, ${I=diag(I_1,\ldots,I_n)}$ with a symmetry $I_1=I_2=\ldots=I_{r} \ne I_{r+1}=I_{r+2}=\ldots=I_n$. It is shown that general trajectories
are quasiperiodic, while for $r\ne 1$, $n1$ the Chaplygin reducing multiplier method does not apply.

Golousov S. V., Khusainov R. R., Savin S. I.
Abstract
The paper deals with one of the modern challenges in walking robotics: moving across
a rough terrain where the geometry of the terrain is unknown and hence it is impossible to plan
precise trajectories for the robot feet in advance, before a collision with the supporting surface
occurs. In this paper, an algorithm for the dynamics correction of the foot trajectory based on
the compliant control is employed to deal with the problem. Additionally, to solve the problem
of dynamic correction of the foot trajectory, it also provides a biomorphic reaction force profile,
which might be a desired property for some applications.

Karavaev Y. L., Shestakov V. A., Yefremov K. S.
Abstract
This paper presents experimental investigations of the control algorithm of a highly maneuverable
mobile manipulation robot. The kinematics of a mobile manipulation robot, the
algorithm of trajectory planning of the mobile robot to the point of object gripping are considered.
By realization of the algorithm, the following tasks are solved: solution of the inverse
positional task for the mobile manipulation robot; motion planning of the mobile manipulator
taking into account the minimization of energy and time consumption per movement. The result
of the algorithm is a movement to the point of gripping of the manipulation object; grasping
and loading of the object. Experimental investigations of the developed algorithms are given.

Karavaev Y. L., Kilin A. A.
Abstract
This paper presents the results of the study of the dynamics of a real spherical robot of
combined type in the case of control using small periodic oscillations. The spherical robot is set
in motion by controlled change of the position of the center of mass and by generating variable
gyrostatic momentum. We demonstrate how to use small periodic controls for stabilization of
the spherical robot during motion. The results of numerical simulation are obtained for various
initial conditions and control parameters that ensure a change in the position of the center of
mass and a variation of gyrostatic momentum. The problem of the motion of a spherical robot
of combined type on a surface that performs flat periodic oscillations is also considered. The
results of numerical simulation are obtained for different initial conditions, control actions and
parameters of oscillations.

Kholostova O. V.
Abstract
The motion of a solid (satellite) carrying a moving point mass in the central Newtonian
gravitational field in an elliptical orbit of arbitrary eccentricity is considered. The law of motion
of a point mass is assumed to allow for the existence of relative equilibria of the “bodypoint”
system in the orbital coordinate system. A nonlinear stability analysis of these equilibria is carried
out, based on the construction and normalization of the areapreserving mapping generated
by the motions of the system.

Kozlov V. V.
Abstract
It is well known that the maximal value of the central moment of inertia of a closed homogeneous
thread of fixed length is achieved on a curve in the form of a circle. This isoperimetric
property plays a key role in investigating the stability of stationary motions of a flexible thread.
A discrete variant of the isoperimetric inequality, when the mass of the thread is concentrated
in a finite number of material particles, is established. An analog of the isoperimetric inequality
for an inhomogeneous thread is proved.

Kurakin L. G., Kurdoglyan A. V.
Abstract
The systems of differential equations with one cosymmetry are considered [1]. The ordinary
object for such systems is a onedimensional continuous family of equilibria. The stability spectrum
changes along this family, but it necessarily contains zero. We consider the nondegeneracy
condition, thus the boundary equilibria separate the family on linearly stable and instable areas.
The stability of the boundary equilibria depends on nonlinear terms of the system. The stability problem for the systems with one cosymmetry is studied in [2]. The general problem is to apply the stability criteria one needs to compute coefficients of the model system. It is especially difficult if the system has a large dimension, while a number of critical variables may be small. A method for calculating coefficients is proposed in [3]. In this work the expressions for the known stability criteria are proposed in a form convenient for calculation. The explicit formulas of the coefficients of the model system are given in semiinvariant form. They are expressed using the generalized eigenvectors of the linear matrix and its conjugate matrix. 
Kurakin L. G., Ostrovskaya I. V.
Abstract
In this paper the twolayer geostrophic model of the rotating fluid and the model of Bessel vortices are considered. Kirchhoff's model of vortices in a homogeneous fluid is the limiting case of both of these models. Part of the study is performed for an arbitrary Hamiltonian depending on the distances between point vortices. The review of the stability problem of stationary rotation of regular Thomson's vortex $N$gon of identical vortices is given for ${N\geqslant 2}$. The stability problem of the vortex tripole/quadrupole is also considered. This axisymmetric vortex structure consists of a~central vortex of an arbitrary intensity and two/three identical peripheral vortices. In the model of a twolayer fluid, peripheral vortices belong to one of the layers and the central vortex can belong to either another layer or the same. The stability of the stationary rotation is interpreted as orbital stability (the stability of a oneparameter orbit of a stationary rotation of a vortex system). The instability of the stationary rotation is instability of equilibrium of the reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The parameter space is divided into three parts: $\bf A$ is the domain of stability in an exact nonlinear setting, $\bf B$ is the linear stability domain, where the stability problem requires nonlinear analysis, and $\bf C$ is the instability domain. In the stability problem of a vortex multipole, another definition of stability is used; it is the stability of an invariant threeparametric set of all trajectories of the families of stationary orbits. It is shown that in the case of non zero total intensity, the stability of the invariant set implies orbital stability. 
Kuzenov V. V., Ryzhkov S. V.
Abstract
A statement of the problem is presented and numerical modeling of plasmagasdynamic
processes in the capillary discharge plume is performed. In the developed model, plasma dynamic
processes in a capillary discharge are determined by the intensity, duration of plasma formation
processes in the capillary discharge channel, and thermodynamic parameters in the surrounding
gaseous medium. The spatial distribution of temperature, density and pressure, radial and
longitudinal velocities of pulsed jets of several capillary discharge channels is presented.

Kuznetsov S. P.
Abstract
The article considers the Chaplygin sleigh on a plane in a potential well, assuming that
an external potential force is supplied at the mass center. Two particular cases are studied in
some detail, namely, a onedimensional potential valley and a potential with rotational symmetry;
in both cases the models reduce to fourdimensional differential equations conserving
mechanical energy. Assuming the potential functions to be quadratic, various behaviors are observed
numerically depending on the energy, from those characteristic to conservative dynamics
(regularity islands and chaotic sea) to strange attractors. This is another example of a nonholonomic
system manifesting these phenomena (similar to those for Celtic stone or Chaplygin
top), which reflects a fundamental nature of these systems occupying an intermediate position
between conservative and dissipative dynamics.

Mashtakov A. P.
Abstract
This paper summarizes results of a sequence of works related to usage of subRiemannian
(SR) geometry in image processing and modeling of the human visual system. In recent research
in psychology of vision (J. Petitot, G.Citti, A. Sarti) it was shown that SR geodesics appear as
natural curves that model a mechanism of the primary visual cortex V1 of a human brain for
completion of contours that are partially corrupted or hidden from observation. We extend the
model to include data adaptivity via a suitable external cost in the SR metric. We show that
data adaptive SR geodesics are useful in real image analysis applications and provide a refined
model of V1 that takes into account the presence of a visual stimulus.

Podobryaev A.
Abstract
We consider leftinvariant optimal control problems on connected Lie groups. We describe
the symmetries of the exponential map that are induced by the symmetries of the vertical part
of the Hamiltonian system of the Pontryagin maximum principle. These symmetries play a key
role in investigation of optimality of extremal trajectories. For connected Lie groups such that
the generic coadjoint orbit has codimension not more than 1 and a connected stabilizer we
introduce a general construction for such symmetries of the exponential map.

Sachkov Y. L., Sachkova E. F.
Abstract
The leftinvariant subRiemannian problem with the growth vector (2, 3, 5, 8) is considered.
A twoparameter group of infinitesimal symmetries consisting of rotations and dilations
is described. The abnormal geodesic flow is factorized modulo the group of symmetries. A parameterization
of the vertical part of abnormal geodesic flow is obtained.

Sakharov A. V.
Abstract
The problem of threedimensional motion of a passively gravitating point in the potential
created by a homogeneous thin fixed ring and a point located in the center of the ring is
considered. Motion of the point allows two first integrals. In the paper equilibrium points and
invariant manifolds of the phase space of the system are found. Motions in them are analyzed.
Bifurcations in the phase plane corresponding to the motion in the equatorial plane are shown.

Sukhov E. A.
Abstract
We deal with motions of a dynamically symmetric rigidbody satellite about its center of mass in a central Newtonian gravitational field. In this case the equations of motion possess particular solutions representing the socalled regular precessions: cylindrical, conical and hyperboloidal precession. If a regular precession is stable there exist two types of periodic motions in its neighborhood: shortperiodic motions with a period close to $2\pi / \omega_2$ and longperiodic motions with a~period close to $2 \pi / \omega_1$ where $\omega_2$ and $\omega_1$ are the frequencies of the linearized system ($\omega_2 > \omega_1$). In this work we obtain analytically and numerically families of shortperiodic motions arising from regular precessions of a symmetric satellite in a nonresonant case and longperiodic motions arising from hyperboloidal precession in cases of third and fourthorder resonances. We investigate the bifurcation problem for these families of periodic motions and present the results in the form of bifurcation diagrams and Poincaré maps. 
Tafrishi S. A., Bai Y., Svinin M., Esmaeilzadeh E., Yamamoto M.
Abstract
In this paper, the resttorest motion planning problem of a fluidactuated spherical robot
is studied. The robot is driven by moving a spherical mass within a circular fluidfilled pipe
fixed internally to the spherical shell. A mathematical model of the robot is established and
two inverse dynamicsbased feedforward control methods are proposed. They parameterize the
motion of the outer shell or the internal moving mass as weighted Beta functions. The feasibility
of the proposed feedforward control schemes is verified under simulations.

Vorochaeva (Volkova) L. Y., Savin S. I., Yatsun A. S.
Abstract
This paper deals with the gait of a robot crawling on a horizontal rough surface with
dry friction. A distinctive feature of the device is the presence of two supporting elements with
a controlled coefficient of friction, which allows for alternately fixing the supports on the surface.
As a result of numerical simulation, the patterns (laws) of influence on the motion characteristics
of the massdimensional and control parameters of the robot, as well as the parameters of the
supporting surface, are obtained, so that it is possible to find application in the design of the
specified devices.
