Vol. 16, no. 1
Vol. 16, no. 1, 2020
Kurakin L. G., Lysenko I. A.
Abstract
The motion of the system of $N$ point vortices with identical intensity $\Gamma$ in the Alfven model of a twofluid plasma is considered.
The stability of the stationary rotation of $N$ identical vortices disposed uniformly on a circle with radius $R$ is studied for $N = 2,\ldots,5$. This problem has three parameters: the discrete parameter $N$ and two continuous parameters $R$ and $c$, where $c>0$ is the value characterizing the plasma.
Two different definitions of the stability are used  the orbital stability and the stability of a threedimensional invariant set founded by the orbits of a continuous family of stationary rotations. Instability is interpreted as instability of equilibrium of the reduced system. An analytical analysis of eigenvalues of the linearization matrix and the quadratic part of the Hamiltonian is given.
As a result, the parameter space $(N,R,c)$ of this problem for two stability definitions considered is divided into three parts: the domain $\boldsymbol{A}$ of stability in an exact nonlinear problem setting, the linear stability domain $\boldsymbol{B}$, where the nonlinear analysis is needed, and the domain of exponential instability $\boldsymbol{C}$. The application of the stability theory of invariant sets for the systems with a few integrals for $N=2,3,4$ allows one to obtain new statements about the stability in the domains, where nonlinear analysis is needed in investigating the orbital stability.

Kuznetsov S. P.
Abstract
Examples of onedimensional lattice systems are considered, in which patterns of different
spatial scales arise alternately, so that the spatial phase over a full cycle undergoes transformation
according to an expanding circle map that implies the occurrence of Smale–Williams attractors
in the multidimensional state space. These models can serve as a basis for design electronic
generators of robust chaos within a paradigm of coupled cellular networks. One of the examples
is a mechanical pendulum system interesting and demonstrative for research and educational
experimental studies.

Kashchenko D. S., Kashchenko S. A.
Abstract
This paper studies the dynamics of a system of two coupled selfexcited oscillators of first
order with onoff delayed feedback using numerical and analytical methods. Regions of “fast” and
“long” synchronization are identified in the parameter space, and the problem of synchronization
on an unstable cycle is examined. For small coupling coefficients it is shown by analytical
methods that the dynamics of the initial system is determined by the dynamics of a special
onedimensional map.

Kuzenov V. V., Shumaev V. V.
Abstract
One of the main obstacles to the uniform laser compression of a fusion target is the plasma
formation instability (the Rayleigh –Taylor instability is the most dangerous). In all the schemes
considered, the impulsive character is important. In this case, not all possible plasma instabilities
are dangerous, but only those that most rapidly increase with time (for example, Rayleigh –
Taylor instability).

Kuznetsov S. P., Kruglov V. P., Sedova Y. V.
Abstract
We discuss two mechanical systems with hyperbolic chaotic attractors of Smale – Williams
type. Both models are based on Froude pendulums. The first system is composed of two
coupled Froude pendulums with alternating periodic braking. The second system is Froude
pendulum with timedelayed feedback and periodic braking. We demonstrate by means of
numerical simulations that the proposed models have chaotic attractors of Smale – Williams
type. We specify regions of parameter values at which the dynamics corresponds to the Smale –
Williams solenoid. We check numerically the hyperbolicity of the attractors.

Kholostova O. V.
Abstract
The motion of a heavy rigid body with a mass geometry corresponding to the Hess case is
considered. The suspension point of the body is assumed to perform highfrequency periodic
vibrations of small amplitude in the threedimensional space. It is proved that for any law of
vibrations of this type, the approximate autonomous equations of the body motion admit an
invariant relation (the first integral at the zero level), which coincides with a similar relation that
exists in the Hess case of the motion of a body with a fixed point. In the approximate equations
of motion written in Hamiltonian form, the cyclic coordinate is introduced and the corresponding
reduction is performed. For the laws of vibration of the suspension point corresponding to the
integrable cases (when there is another cyclic coordinate in the system), a detailed study of
the model onedegreeoffreedom system is given. For the nonintegrable cases, an analogy with
the approximate problem of the motion of a Lagrange top with a vibrating suspension point is
drawn, and the results obtained earlier for the top are used. Some properties of the body motion
at the nonzero level of the above invariant relation are also discussed.

Hart'anský R., Hricko J., Mierka M., Halgoš J., Dzuriš M.
Abstract
The paper is focused on the design of a compact compliant mechanical body (CCMB) that
will be used as a transducer of the oneaxis force to the distance of two plates. The conversion
principle of the distance (which carries information about the acting force) to the frequency of
the electrical signal is carefully described in the article too.

Shaitor N. M., Yakimovich B. A., Ryaskov Y. I., Gorpinchenko A. V.
Abstract
The current trends in the creation of underwater robotic devices for the study of the world
ocean are considered. It is noted that complex types of motion of various effectors cannot be
implemented by means of simple electromechanical transducers. Therefore, the use of genetic
engineering techniques to create a variety of electromechanical structures that can provide effective
functionality of marine robotics objects solves a scientific problem that is of much current
interest. The aim of this work is to analyze and study the possibility of applying the principles of genetic synthesis for the development of new types of energyefficient transducers built on the basis of complex electromechanical structures and demanded by a wide range of the developing marine robotics. It is shown how complex electromechanical structures that satisfy a given target function are obtained using genetic operators, together with operators of geometric transformations. The use of genetic engineering techniques for the development of electromechanical transducers expands the possibilities of creating and studying the structural diversity of mechanisms with complex movement of effectors. The configuration of a wave fin propulsion device based on the inductor electromechanical transducer of axialradial configuration is considered. A mathematical model, methods and a program for calculating an electromechanical transducer with complex structural axialradial configuration is developed. The problem of determining the optimal copper/steel ratio is solved, the leading dimensions are obtained and the rational relations of the parameters for calculation of the axialradial configuration transducer with a minimum mass are established. Calculations show that, in comparison with the traditional inductor machines with drumtype rotors, the mass of the active electromagnetic core of axialradial type transducers can be reduced by a factor of 2.7 to 3.4 with a double reduction in volume. Axialradial configurations of electromechanical structures designed for electric frequencies of 50 to 400 Hz are suitable for use in lowspeed drives of wind and hydroelectric power plants and propulsion systems of underwater vehicles, and those designed for 1000 to 10 000 Hz can be used in highspeed drives of autonomous power plants of vehicles, aircraft, gyromotors, gas and steam turbines. 
Vorochaeva (Volkova) L. Y., Savin S. I., Malchikov A. V.
Abstract
This paper approaches the issue of the jumping robot overcoming one step of a flight of
stairs. A classification of obstacles according to the way they are surmounted is proposed, the
basic concepts concerning the flight of stairs and the realization of a leap from one step to
another are introduced. A numerical simulation of a robot’s jump, carried out with a certain
initial velocity, the vector of which is located at a certain angle to the horizon, has been carried
out. The influence on the ranges of these two values of the numerical values of the height and
the length of the step, the ratios between the length and the height of the step, as well as the
distance to the step from which the jump is performed have been established.

Andreev A. S., Peregudova O. A.
Abstract
This paper addresses the trajectory tracking control design of an omnidirectional mobile
robot with a center of mass displaced from the geometrical center of the robot platform. Due
to the high maneuverability provided by omniwheels, such robots are widely used in industry
to transport loads in narrow spaces. As a rule, the center of mass of the load does not coincide
with the geometric center of the robot platform. This makes the trajectory tracking control
problem of a robot with a displaced center of mass relevant. In this paper, two controllers
are constructed that solve the problem of global trajectory tracking control of the robot. The
controllers are designed based on the Lyapunov function method. The main difficulty in applying
the Lyapunov function method for the trajectory tracking control problem of the robot is that
the time derivative of the Lyapunov function is not definite negative, but only semidefinite
negative. Moreover, the LaSalle invariance principle is not applicable in this case since the
closedloop system is a nonautonomous system of differential equations. In this paper, it is
shown that the quasiinvariance principle for nonautonomous systems of differential equations
is much convenient for the asymptotic stability analysis of the closedloop system. Firstly,
we construct an unbounded state feedback controller such as proportionalderivative term plus
feedforward. As a result, the global uniform asymptotic stability property of the origin of
the closedloop system has been proved. Secondly, we find that, if the damping forces of the
robot are large enough, then the saturated position output feedback controller solves the global
trajectory tracking control problem without velocity measurements. The effectiveness of the
proposed controllers has been verified through simulation tests. Namely, a comparative analysis
of the bounded controller obtained and the wellknown “PD+” like control scheme is carried
out. It is shown that the approach proposed in this paper saves energy for control inputs.
Besides, a comparative analysis of the bounded controller and its analogue constructed earlier
in a cylindrical phase space is carried out. It is shown that the controller provides lower values
for the root mean square error of the position and velocity of the closedloop system.

de la Cruz M., Gaspar N., Linares R.
Abstract
In this paper we revisit the construction by which the $SL(2,\mathbb{R})$ symmetry of the Euler equations
allows a simple pendulum to be obtained from a rigid body. We begin by reviewing the
original relation found by Holm and
Marsden in which, starting from the two integrals of motion of the extended rigid body with Lie algebra
$\mathfrak{iso}(2)$ and introducing a proper momentum map, it is possible to obtain
both the Hamiltonian and the equations of motion of the pendulum. Important in this
construction is the fact that both integrals of motion have the
geometry of an elliptic cylinder. By considering the whole $SL(2,\mathbb{R})$ symmetry
group, in this contribution we
give all possible combinations of the integrals of motion and the corresponding
momentum maps that produce the
simple pendulum, showing that this system can also appear when the geometry of one
of the integrals of motion is
given by a hyperbolic cylinder and the other by an elliptic cylinder. As a result,
we show that, from the extended
rigid body with Lie algebra~$\mathfrak{iso}(1,1)$, it is possible to obtain the
pendulum, but only in circulating movement.
Finally, as a byproduct of our analysis we provide the momentum maps that give
origin to the pendulum with an
imaginary time. Our discussion covers both the algebraic and the geometric point
of view.

Hricko J., Havlík Š., Karavaev Y. L.
Abstract
The paper is focused to design, simulation and modeling of the compact compliant structures
widely used in construction of robotic devices. As the illustrative example it is proposed
mechanism for reduction of motion, which enables to improve the accuracy of the positioning
system. The physical model is fabricated by 3D printing technology. Its proposed performance
characteristics are verified by measurement on the experimental test bed by using laser distance
sensors and image sensing/processing technology.

Sukhanov V.
Abstract
In this paper, we study the asymptotic behavior of the solutions of the Cauchy problem
for a nonlinear KdV type system associated with the Schrödinger spectral operator with an
energydependent potential. Using the set of motion integrals for this system, we determine the
amplitude of the asymptotic solution in terms of spectral data for the initial condition of the
Cauchy problem.

Shekutkovski N., Shoptrajanov M.
Abstract
This paper concerns the connection between shape theory and attractors for semidynamical
systems in metric spaces. We show that intrinsic shape theory from [6] is a convenient framework
to study the global properties which the attractor inherits from the phase space. Namely,
following [6] we’ll improve some of the previous results about the shape of global attractors in
arbitrary metrizable spaces by using the intrinsic approach to shape which combines continuity
up to a covering and the corresponding homotopies of first order.

Mashtakov A. P., Popov A. Y.
Abstract
We consider a subRiemannian problem on the group of motions of threedimensional space. Such a problem is encountered in the analysis of 3D images as well as in describing the motion of a solid body in a fluid.
Mathematically, this problem reduces to solving a Hamiltonian system the vertical part of which is a system of six differential equations with unknown functions $u_1, \ldots, u_6$.
The optimality consideration arising from the Pontryagin maximum principle implies that the last component of the vector control $\bar{u}$, denoted by $u_6$, must be constant. In the problem of the motion of a solid body
in a fluid, this means that the fluid flow has a unique velocity potential, i.e., is vortexfree.
The case ($u_6 = 0$), which is the most important for applications and at the same time the simplest, was rigorously studied by the authors in 2017. There, a solution to the system
was found in explicit form. Namely, the extremal controls $u_1, \ldots, u_5$ were expressed in terms of elliptic functions. Now we consider the general case: $u_6$ is
an arbitrary constant. In this case, we obtain a solution to the system in an operator form. Although the explicit form of the extremal controls does not follow directly from these
formulas (their calculation requires the inversion of some nontrivial operator), it allows us to construct an approximate analytical solution for a small parameter $u_6$. Computer
simulation shows a good agreement between the constructed analytical approximations and the solutions computed via numerical integration of the system.
