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# 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 two-fluid 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 three-dimensional 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. Keywords: point vortex, two-fluid plasma, stability, stationary rotation, Hamiltonian system, invariant set Citation: Kurakin L. G.,  Lysenko I. A., On the Stability of the Orbit and the Invariant Set of Thomson’s Vortex Polygon in a Two-Fluid Plasma, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 3-11 DOI:10.20537/nd200101
 Kuznetsov S. P. Abstract Examples of one-dimensional 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. Keywords: dynamical system, chaos, attractor, Smale – Williams solenoid, Turing pattern, pendulum, parametric oscillations, cellular neural network Citation: Kuznetsov S. P., Some Lattice Models with Hyperbolic Chaotic Attractors, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 13-21 DOI:10.20537/nd200102
 Kashchenko D. S.,  Kashchenko S. A. Abstract This paper studies the dynamics of a system of two coupled self-excited oscillators of first order with on-off 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 one-dimensional map. Keywords: stability, dynamics, relaxation cycles, irregular oscillations Citation: Kashchenko D. S.,  Kashchenko S. A., Dynamics of a System of Two Simple Self-Excited Oscillators with Delayed Step-by-Step Feedback, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 23-43 DOI:10.20537/nd200103
 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). Keywords: laser, magnetic field, mathematical model, plasma target Citation: Kuzenov V. V.,  Shumaev V. V., Estimation of Instabilities under the Joint Action of Laser Radiation and a Magnetic Field on a Plasma, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 45-50 DOI:10.20537/nd200104
 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 time-delayed 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. Keywords: hyperbolic chaotic attractors, Smale – Williams solenoid, Bernoulli map Citation: Kuznetsov S. P.,  Kruglov V. P.,  Sedova Y. V., Mechanical Systems with Hyperbolic Chaotic Attractors Based on Froude Pendulums, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 51-58 DOI:10.20537/nd200105
 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 high-frequency periodic vibrations of small amplitude in the three-dimensional 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 one-degree-of-freedom 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. Keywords: Hess case, high-frequency vibrations, integrable case, reduced system, Lagrange top Citation: Kholostova O. V., On the Dynamics of a Rigid Body in the Hess Case at High-Frequency Vibrations of a Suspension Point, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 59-84 DOI:10.20537/nd200106
 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 one-axis 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. Keywords: compact compliant mechanical body, mechanical force sensor, transfer response Citation: Hart'anský R.,  Hricko J.,  Mierka M.,  Halgoš J.,  Dzuriš M., MEMS Sensor of Force, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 85-92 DOI:10.20537/nd200107
 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 energy-efficient 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 axial-radial configuration is considered. A mathematical model, methods and a program for calculating an electromechanical transducer with complex structural axial-radial 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 axial-radial 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 axial-radial type transducers can be reduced by a factor of 2.7 to 3.4 with a double reduction in volume. Axial-radial configurations of electromechanical structures designed for electric frequencies of 50 to 400 Hz are suitable for use in low-speed 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 high-speed drives of autonomous power plants of vehicles, aircraft, gyromotors, gas and steam turbines. Keywords: underwater vehicles, electromechanical structure, genetic procedure, electromagnetic chromosome, axial-radial configuration, wave propulsion device, electromagnetic core, calculation method Citation: Shaitor N. M.,  Yakimovich B. A.,  Ryaskov Y. I.,  Gorpinchenko A. V., Application of Genetic Engineering Techniques in the Development of Complex Electromechanical Structures for Marine Robotics, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 93-103 DOI:10.20537/nd200108
 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. Keywords: jumping robot, obstacle, flight of stairs, separation velocity, step dimensions Citation: Vorochaeva (Volkova) L. Y.,  Savin S. I.,  Malchikov A. V., An Approach to Moving over Obstacles for a Wheeled Jumping Robot, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 105-114 DOI:10.20537/nd200109
 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 closed-loop system is a nonautonomous system of differential equations. In this paper, it is shown that the quasi-invariance principle for nonautonomous systems of differential equations is much convenient for the asymptotic stability analysis of the closed-loop system. Firstly, we construct an unbounded state feedback controller such as proportional-derivative term plus feedforward. As a result, the global uniform asymptotic stability property of the origin of the closed-loop 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 well-known “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 closed-loop system. Keywords: omnidirectional mobile robot, displaced mass center, global trajectory tracking control, output position feedback, Lyapunov function method Citation: Andreev A. S.,  Peregudova O. A., On Global Trajectory Tracking Control for an Omnidirectional Mobile Robot with a Displaced Center of Mass, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 115-131 DOI:10.20537/nd200110
 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. Keywords: free motion of a rigid body, simple pendulum, bi-Hamiltonian structures, momentum maps, symplectic reduction Citation: de la Cruz M.,  Gaspar N.,  Linares R., The Extended Rigid Body and the Pendulum Revisited, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 133-159 DOI:10.20537/nd200111
 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. Keywords: compact compliant mechanisms, 3D printing, modeling and simulation, HIL simulations Received Citation: Hricko J.,  Havlík Š.,  Karavaev Y. L., Verifying the Performance Characteristics of the (micro) Robotic Devices, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 161-172 DOI:10.20537/nd200112
 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 energy-dependent 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. Keywords: nonlinear KdV type system, asymptotic behavior, energy-dependent potential, motion integrals Citation: Sukhanov V., Asymptotic Behavior of Solutions of a System of KdV Type Associated with the Schrödinger Operator with an Energy-Dependent Potential, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 173-179 DOI:10.20537/nd200113
 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. Keywords: intrinsic shape, regular covering, continuity over a covering, attractor, proximate net Citation: Shekutkovski N.,  Shoptrajanov M., Intrinsic Shape Property of Global Attractors in Metrizable Spaces, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 181-194 DOI:10.20537/nd200114
 Mashtakov A. P.,  Popov A. Y. Abstract We consider a sub-Riemannian problem on the group of motions of three-dimensional 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 vortex-free. 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. Keywords: Hamiltonian system, Pontryagin maximum principle, sub-Riemannian, Lie group Received Citation: Mashtakov A. P.,  Popov A. Y., Asymptotics of Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 1, pp. 195-208 DOI:10.20537/nd200115

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