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Boxonov Z. S.
In this paper, we study discrete-time 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.
Keywords: Lyapunov function, fixed point, limit point, invariant set
Citation: Boxonov Z. S., Description of Trajectories of an Evolution Operator Generated by Mosquito Population, Rus. J. Nonlin. Dyn., 2024, Vol. 20, no. 2, pp. 197-207
Gorr G. V.
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 first-order equation with separating variables.
Keywords: three homogeneous force fields, precessional motions, invariant relation
Citation: Gorr G. V., On a Method for Integrating the Equations of Rigid Body Motion in Three Homogeneous Force Fields, Rus. J. Nonlin. Dyn., 2024, Vol. 20, no. 2, pp. 209-218
Zemlyanukhin A. I.,  Bochkarev A. V.,  Artamonov N. A.
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 soliton-like 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.
Keywords: cylindrical shell, solitary wave solution, physically admissible solution, nonlinear deformation law, refined shell model
Citation: Zemlyanukhin A. I.,  Bochkarev A. V.,  Artamonov N. A., Physically Admissible and Inadmissible Exact Localized Solutions in Problems of Nonlinear Wave Dynamics of Cylindrical Shells, Rus. J. Nonlin. Dyn., 2024, Vol. 20, no. 2, pp. 219-229
Piskun N. V.,  Lukin A. V.,  Popov I. A.,  Shtukin L. V.
In this work, we study the nonlinear dynamics of a mode-localized 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 steady-state 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, amplitude-frequency 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.
Keywords: nano- and microsystems, nano- and microelectromechanical systems (NEMS/MEMS), sensors of masses of deposited particles, gas detectors, gas concentration sensors, nonlinear dynamics, coupled-field problems, modal localization, weakly coupled systems
Citation: Piskun N. V.,  Lukin A. V.,  Popov I. A.,  Shtukin L. V., Mass Sensing by Symmetry Breaking and Mode Localization in a System of Parametrically Excited Microbeam Resonators, Rus. J. Nonlin. Dyn., 2024, Vol. 20, no. 2, pp. 231-257
Gudimenko A. I.,  Lihosherstov A. V.
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 low-frequency high-amplitude oscillations was discovered. The primary analysis showed that this phenomenon has much in common with self-oscillations 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 self-oscillations 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.
Keywords: linear dynamical system, harmonic chain, high-amplitude oscillations
Citation: Gudimenko A. I.,  Lihosherstov A. V., On the Phenomenon of Low-Frequency, Large-Amplitude Oscillations in a High-Dimensional Linear Dynamical System, Rus. J. Nonlin. Dyn., 2024, Vol. 20, no. 2, pp. 259-276
Hamidouche B.,  Guesmi K.,  Essounbouli N.
This paper presents a comprehensive investigation of delayed sliding-mode 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.
Keywords: chaotic systems, sliding mode control, synchronization, time-delayed system, unstable periodic orbits, Rössler benchmark
Citation: Hamidouche B.,  Guesmi K.,  Essounbouli N., Delayed Sliding Mode Control of Chaotic Systems, Rus. J. Nonlin. Dyn., 2024, Vol. 20, no. 2, pp. 277-293
Nor N. V.
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 actor-network 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.
Keywords: control, control of a mechanical system, spherical robot, mechanics, artificial intelligence, reinforcement learning, Q-learning, DDPG, actor-critic, multilayer neural network, MuJoCo, PyTorch
Citation: Nor N. V., Reinforcement Learning in the Task of Spherical Robot Motion Control, Rus. J. Nonlin. Dyn., 2024, Vol. 20, no. 2, pp. 295-310
Moiseev G. N.
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 step-by-step algorithm for finding out reaction components in the case of controlled motion. Based on these results, no-overturn 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.
Keywords: omnidirectional vehicle, omniwheel, Mecanum wheel, reaction force, nonholonomic model, unilateral constraints, constraint violation
Citation: Moiseev G. N., No-Overturn Conditions for Omnivehicle Motion, Rus. J. Nonlin. Dyn., 2024, Vol. 20, no. 2, pp. 311-336