 EDITORINCHIEF
 Honorary Editor
 Editorial board

 Passed away
Boxonov Z. S.
Abstract
In this paper, we study discretetime dynamical systems generated by the evolution operator
of mosquito population. An invariant set is found and a Lyapunov function with respect to the
operator is constructed in this set. Using the Lyapunov function, the global attraction of a fixed
point is proved. Moreover, we give some biological interpretations of our results.

Gorr G. V.
Abstract
This paper presents a method for integrating the equations of motion of a rigid body having
a fixed point in three homogeneous force fields. It is assumed that under certain conditions these
equations admit an invariant relation that is characterized by the following property: the velocity
of proper rotation of the body is twice as large as the velocity of precession. The integration of
the initial system is reduced to the study of three algebraic equations for the main variables of
the problem and one differential firstorder equation with separating variables.

Zemlyanukhin A. I., Bochkarev A. V., Artamonov N. A.
Abstract
It is shown that, when studying nonlinear longitudinal deformation waves in cylindrical
shells, it is possible to obtain physically admissible solitary wave solutions using refined shell
models. In the article, a physically admissible exact localized solution based on the Flügge –
Lurie – Byrne model is constructed. An analysis of the influence of the external nonlinear elastic
medium on the exact solutions obtained is carried out. It is established that the use of quadratic
and cubic nonlinear deformation laws leads to nonintegrable equations with exact solitonlike
solutions. However, the amplitudes of the exact solutions exceed the values of permissible displacements
corresponding to the maximum points on the curves of the deformation laws of the
external medium, which leads to the physical inadmissibility of these solutions.

Piskun N. V., Lukin A. V., Popov I. A., Shtukin L. V.
Abstract
In this work, we study the nonlinear dynamics of a modelocalized mass detector. A system of
equations is obtained for two weakly coupled beam resonators with an alternating electric current
flowing through them and taking into account the point mass on one of the resonators. The onedimensional
problem of thermal conductivity is solved, and a steadystate harmonic temperature
distribution in the volume of the resonators is obtained. Using the method of multiple scales,
a system of equations in slow variables is obtained, on the basis of which instability zones
of parametric resonance, amplitudefrequency characteristics, as well as zones of attraction of
various branches, are found. It is shown that in a completely symmetrical system (without
a deposited particle), the effect of branching of the antiphase branch of the frequency response
is observed, which leads to the existence of an oscillation regime with different amplitudes in a
certain frequency range. In the presence of a deposited particle, this effect is enhanced, and the
branching point and the ratio of the amplitudes of oscillations of the resonators depend on the
mass of the deposited particle.

Gudimenko A. I., Lihosherstov A. V.
Abstract
This article considers a linear dynamic system that models a chain of coupled harmonic
oscillators, under special boundary conditions that ensure a balanced energy flow from one end
of the chain to the other. The energy conductivity of the chain is controlled by the parameter $\alpha$
of the system. In a numerical experiment on this system, with a large number of oscillators and at certain values of $\alpha$, the phenomenon of lowfrequency highamplitude oscillations was discovered. The primary analysis showed that this phenomenon has much in common with selfoscillations in nonlinear systems. In both cases, periodic motion is created and maintained by an internal energy source that does not have the corresponding periodicity. In addition, the amplitude of the oscillations significantly exceeds the initial state amplitude. However, this phenomenon also has a fundamental difference from selfoscillations in that it is controlled by the oscillation synchronization mechanism in linear systems and not by the exponential instability suppression mechanism in nonlinear systems. This article provides an explanation of the observed phenomenon on the basis of a complete analytical solution of the system. The solution is constructed in a standard way by reducing the dynamic problem to the problem of eigenvalues and eigenvectors for the system matrix. When solving, we use methods from the theory of orthogonal polynomials. In addition, we discuss two physical interpretations of the system. The connection between these interpretations and the system is established through the Schrödinger variables. 
Hamidouche B., Guesmi K., Essounbouli N.
Abstract
This paper presents a comprehensive investigation of delayed slidingmode control and synchronization
of chaotic systems. The findings of this paper offer valuable insights into chaos
control and synchronization and provide promising prospects for practical applications in various
domains where the control of complex dynamical systems is a critical question. In this paper,
we propose three approaches of control to regulate chaotic behavior and induce synchronization
between the system’s state and its delayed value, by one period, of the unstable periodic orbits
(UPOs). The stabilization ability of each controller is demonstrated analytically based on Lyapunov
theory. Furthermore, we provide a bridge between classical stability and structural one
through the use of the synchronization error, as an argument of the controller, instead of the
classical tracking error. Through three sets of simulations, we demonstrate the effectiveness of the proposed approaches in driving the chaotic system toward stable, simple, and predictable periodic behavior. The results confirm the rapid achievement of stabilization, even with changes in the sliding surface and control activation time point showing, hence, the approaches’ adaptability and reliability. Furthermore, the controlled system exhibits remarkable insensitivity to changes in initial conditions, thus showing the robustness of the proposed control strategies. 
Nor N. V.
Abstract
This article discusses one of the DDPG (Deep Deterministic Policy Gradient) reinforcement
learning algorithms applied to the problem of motion control of a spherical robot. Inside the
spherical robot shell there is a platform with a wheel, and the robot is simulated in the MuJoCo
physical simulation environment. The goal is to teach the robot to move along an arbitrary closed curve with minimal error. The output control algorithm is a pair of trained neural networks — actor and critic, where the actornetwork is used to obtain the control torques applied to the robot wheel and the criticnetwork is only involved in the learning process. The results of the training are shown below, namely how the robot performs the motion along ten arbitrary trajectories, where the main quality functional is the average error magnitude over the trajectory length scale. The algorithm is implemented using the PyTorch machine learning library. 
Moiseev G. N.
Abstract
We consider the dynamics of an omnidirectional vehicle moving on a perfectly rough horizontal
plane. The vehicle has three omniwheels controlled by three direct current motors. To find out the limits of the scope where the bilateral constraints model is applicable, we study the normal reactions of the vehicle. We present a stepbystep algorithm for finding out reaction components in the case of controlled motion. Based on these results, nooverturn conditions of the vehicle are proposed. We apply this approach to study a specific model, that of a symmetrical omnivehicle. As a consequence, vehicle design recommendations are proposed. 