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Berestova S. A.,  Prosviryakov E. Y.
An exact solution of the Oberbeck – Boussinesq equations for the description of the steadystate Bénard – Rayleigh convection in an infinitely extensive horizontal layer is presented. This exact solution describes the large-scale motion of a vertical vortex flow outside the field of the Coriolis force. The large-scale fluid flow is considered in the approximation of a thin layer with nondeformable (flat) boundaries. This assumption allows us to describe the large-scale fluid motion as shear motion. Two velocity vector components, called horizontal components, are taken into account. Consequently, the third component of the velocity vector (the vertical velocity) is zero. The shear flow of the vertical vortex flow is described by linear forms from the horizontal coordinates for velocity, temperature and pressure fields. The topology of the steady flow of a viscous incompressible fluid is defined by coefficients of linear forms which have a dependence on the vertical (transverse) coordinate. The functions unknown in advance are exactly defined from the system of ordinary differential equations of order fifteen. The coefficients of the forms are polynomials. The spectral properties of the polynomials in the domain of definition of the solution are investigated. The analysis of distribution of the zeroes of hydrodynamical fields has allowed a definition of the stratification of the physical fields. The paper presents a detailed study of the existence of steady reverse flows in the convective fluid flow of Bénard – Rayleigh – Couette type.
Keywords: exact solution, shear flow, inhomogeneous flow, convection, Oberbeck – Boussinesq system, class of Lin – Sidorov – Aristov solutions, vertical swirl of fluid, reverse flow, stratification
Citation: Berestova S. A.,  Prosviryakov E. Y., An Inhomogeneous Steady-State Convection of a Vertical Vortex Fluid, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp. 167-186
Rozenblat G. M.,  Grishakin V. T.
This paper deals with a formulation and a solution of problems of the dynamics of mechanical systems for which solutions that do not take into account the unilateral nature of the constraints imposed on the objects under study have been obtained before. The motive force in all the cases considered is the gravity force applied to the center of mass of each body of the mechanical system. Since unilateral constraints are imposed on all systems of bodies considered in the abovementioned problems, their correct solution requires taking into account the unilateral action of the constraint reaction forces applied to the bodies of the systems under study. A detailed analysis of the motion of the systems after zeroing out the constraint reaction forces is carried out. Results of numerical experiments are presented which are used to construct motion patterns of the systems of bodies illustrating the motions of the above-mentioned systems after they lose contact with the supporting surfaces.
Keywords: unilateral constraints, normal reactions, kinetic energy, free fall
Citation: Rozenblat G. M.,  Grishakin V. T., Singularities of the Dynamics of Some Bar Systems with Unilateral Constraints, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp. 187-200
Ledkov A.,  Pikalov R.
Tether retrieval is an important stage in many projects using space tether systems. It is known that uniform retrieval is an unstable process that leads to the winding of the tether on a satellite at the final stage of retraction. This is a serious obstacle to the practical application of space tethers in the tasks of climbing payloads to a satellite and docking the spacecraft with a tethered satellite after its capture. The paper investigates the plane motion of a space tether system with a massless tether of variable length in an elliptical orbit. A new control law that ensures the retrieval of the tether without increasing the amplitude of oscillations at the final stage is proposed. The asymptotic stability of the space tether system’s controlled motion in an elliptical orbit is proved. A numerical analysis of tether retrieval is carried out. The influence of the eccentricity of the orbit on the retrieval process is investigated. The results of the work can be useful in preparing missions of the active space debris removal and in performing operations involving tether retrieval.
Keywords: space tether system, retrieval, nonlinear, control law, tether
Citation: Ledkov A.,  Pikalov R., Nonlinear Control of Tether Retrieval in an Elliptical Orbit, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp. 201-218
Shvarts K. G.
In this paper a new exact solution of the Navier – Stokes equations in the Boussinesq approximation describing advective flow in a horizontal liquid layer with free boundaries, where the vertical velocity component is a constant value, is obtained. The temperature is linear along the boundaries of the layer. Solutions of this kind are used to close three-dimensional equations averaged across the layer in the derivation of two-dimensional models of nonisothermal large-scale flows in a thin layer of liquid or incompressible gas. The properties of advective flow at different values of Reynolds number and Prandtl number are investigated.
Keywords: advective flow, exact solution, Navier – Stokes equation
Citation: Shvarts K. G., Plane-Parallel Advective Flow in a Horizontal Layer of Incompressible Permeable Fluid, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp. 219-226
Medvedev V. S.,  Zhuzhoma E. V.
We prove that, given any $n\geqslant 3$ and $2\leqslant q\leqslant n-1$, there is a closed $n$-manifold $M^n$ admitting a chaotic lamination of codimension $q$ whose support has the topological dimension ${n-q+1}$. For $n=3$ and $q=2$, such chaotic laminations can be represented as nontrivial 2-dimensional basic sets of axiom A flows on 3-manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2-dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repeller-attractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with one-dimensional basic sets obtained previously by V. Grines, R. Plykin and Yu. Zhirov [16, 28, 31].
Keywords: chaotic lamination, basic set, axiom A flow
Citation: Medvedev V. S.,  Zhuzhoma E. V., On a Classification of Chaotic Laminations which are Nontrivial Basic Sets of Axiom A Flows, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp. 227-237
Soga K.
This paper provides a quite simple method of Tonelli’s calculus of variations with positive definite and superlinear Lagrangians. The result complements the classical literature of calculus of variations before Tonelli’s modern approach. Inspired by Euler’s spirit, the proposed method employs finite-dimensional approximation of the exact action functional, whose minimizer is easily found as a solution of Euler’s discretization of the exact Euler – Lagrange equation. The Euler – Cauchy polygonal line generated by the approximate minimizer converges to an exact smooth minimizing curve. This framework yields an elementary proof of the existence and regularity of minimizers within the family of smooth curves and hence, with a minor additional step, within the family of Lipschitz curves, without using modern functional analysis on absolutely continuous curves and lower semicontinuity of action functionals.
Keywords: Tonelli’s calculus of variations, direct method, action minimizing, minimizing curve, regularity of minimizer, Euler method, Euler – Cauchy polygon
Citation: Soga K., A Remark on Tonelli’s Calculus of Variations, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp. 239-248
Gorr G. V.
This paper is concerned with a special class of precessions of a rigid body having a fixed point in a force field which is a superposition of three homogeneous force fields. It is assumed that the velocity of proper rotation of the body is twice as large as its velocity of precession. The conditions for the existence of the precessions under study are written in the form of a system of algebraic equations for the parameters of the problem. Its solvability is proved for a dynamically symmetric body. It is proved that, if the ellipsoid of inertia of the body is a sphere, then the nutation angle is equal to $\arccos \frac{1}{3}$. The resulting solution of the equations of motion of the body is represented as elliptic Jacobi functions.
Keywords: three homogeneous force fields, precessions, dynamically symmetric bodies, elliptic functions
Citation: Gorr G. V., On a Class of Precessions of a Rigid Body with a Fixed Point under the Action of Forces of Three Homogeneous Force Fields, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp. 249-264
Klekovkin A. V.,  Karavaev Y. L.,  Mamaev I. S.
This paper presents the design of an aquatic robot actuated by one internal rotor. The robot body has a cylindrical form with a base in the form of a symmetric airfoil with a sharp edge. For this object, equations of motion are presented in the form of Kirchhoff equations for rigid body motion in an ideal fluid, which are supplemented with viscous resistance terms. A prototype of the aquatic robot with an internal rotor is developed. Using this prototype, experimental investigations of motion in a fluid are carried out.
Keywords: mobile robot, aquatic robot, motion simulation
Citation: Klekovkin A. V.,  Karavaev Y. L.,  Mamaev I. S., The Control of an Aquatic Robot by a Periodic Rotation of the Internal Flywheel, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp. 265-279
Moskvitin V. M.,  Semenova N. I.
In recent years, more and more researchers in the field of artificial neural networks have been interested in creating hardware implementations where neurons and the connection between them are realized physically. Such networks solve the problem of scaling and increase the speed of obtaining and processing information, but they can be affected by internal noise.
In this paper we analyze an echo state neural network (ESN) in the presence of uncorrelated additive and multiplicative white Gaussian noise. Here we consider the case where artificial neurons have a linear activation function with different slope coefficients. We consider the influence of the input signal, memory and connection matrices on the accumulation of noise. We have found that the general view of variance and the signal-to-noise ratio of the ESN output signal is similar to only one neuron. The noise is less accumulated in ESN with a diagonal reservoir connection matrix with a large “blurring” coefficient. This is especially true of uncorrelated multiplicative noise.
Keywords: artificial neural networks, recurrent neural network, echo state network, noise, dispersion, statistic, white gaussian noise
Citation: Moskvitin V. M.,  Semenova N. I., Noise Impact on a Recurrent Neural Network with a Linear Activation Function, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp. 281-293