Vol. 14, no. 1
Vol. 14, no. 1, 2018
Skazkina V., Kiselev A. R., Borovkova E. I., Ponomarenko V. I., Prokhorov M. D., Karavaev A. S.
Abstract
The previously proposed method for quantifying the degree of synchronization between circulatory regulation loops is used to analyze the time realizations of healthy subjects. Statistical properties of the index are studied in the analysis of twohour records of experimental signals. In the course of this work, we investigated the properties of the estimation of the degree of synchronization using temporal realizations with different length, and we investigated the features of synchronization between the control loops under study at a time equal to hundreds of characteristic periods.

Kulakov M. P., Frisman E. Y.
Abstract
The article is devoted to the model of spatialtemporal dynamics of agestructured populations coupled by migration. The dynamics of a single population is described by a twodimensional nonlinear map demonstrating multistability, and a coupling is a nonlocal migration of individuals. An analysis is made of the problem of synchronization (complete, cluster and chaotic), chimera states formation and transitions between different types of dynamics. The problem of dependence of the spacetime regimes on the initial states is discussed in detail. Two types of initial conditions are considered: random and nonrandom (special, as defined ratios) and two cases of single oscillator dynamics — regular and irregular fluctuations. A new cluster synchronization mechanism is found which is caused by the multistability of the local oscillator (population), when different clusters differ fundamentally in the type of their dynamics. It is found that nonrandom initial conditions, even for subcritical parameters, lead to complex regimes including various chimeras. A description is given of the spacetime regime when there are several single nonsynchronous elements with large amplitude in a cluster with regular or chaotic dynamics. It is found that the type of spatialtemporal dynamics depends considerably on the distribution parameters of random initial conditions. For a large scale factor and any coupling parameters, there are no coherent regimes at all, and coherent states are possible only for a small scale factor.

Vasiukova O. E., Klimina L. A.
Abstract
The paper presents a mathematical model of a controlled pendulum under the assumption that friction in a joint is a sum of Coulomb and viscous friction. Moreover, it is taken into account that the Coulomb friction torque depends on the value of normal reaction force in a joint. The control torque is chosen as a function that depends only on the sign of the angular speed of the pendulum. Via the Pontryagin approach for nearHamiltonian systems, the program law is constructed for test selfoscillations. Test selfoscillations are to be used for identification of friction coefficients. Bifurcation diagrams are constructed that describe the dependence between amplitudes of selfoscillations and values of the control torque. The proposed approach to the identification of parameters of the friction requires information about amplitudes of test selfoscillations but does not require information about the trajectory of motion as a function of time. Numerical simulation of the motion of the system is carried out. The range of parameter values is described for which the method proposed in the paper is quite accurate.

Burov A. A., Guerman A., Raspopova E., Nikonov V.
Abstract
It is well known that several small celestial objects are of irregular shape. In particular, there exist asteroids of the socalled “dogbone” shape. It turns out that approximation of these bodies by dumbbells, as proposed by V.V. Beletsky, provides an effective tool for analytical investigation of dynamics in vicinities of such bodies. There remains the question of how to divide reasonably a “dogbone” body into two parts using available measurement data. In this paper we introduce an approach based on the socalled $K$mean algorithm proposed by the prominent Polish mathematician H. Steinhaus. 
Fimin N. N., Chechetkin V. M.
Abstract
The properties of quasilinear differential equations with the same the principal part are considered. Their connection with the reduced system of Euler equations is established, which results from the hydrodynamic substitution in the kinetic Liouville and Vlasov equations. When considering the momentum equation of the Euler system, it turns out that it reduces to a special form such as Liouville – Jacobi equation. This equation can also be investigated using a hydrodynamic substitution, but of conjugate type. The application of this substitution (of the second order) makes it possible to symmetrize the technique of applying hydrodynamic substitution and to extend the class of equations of hydrodynamic type to which systems of (in the general case nonHamiltonian) firstorder autonomous differential equations. Examples are given of the use of the developed formalism for systems of gravitating particles in postNewtonian approximation and the hydrodynamic systems described by Monge potentials, with the aim of constructing the Liouville – Jacobi equations and applying to them a modified hydrodynamic substitution.

Georgievsky D. V.
Abstract
This paper is concerned with obtaining the parameters of a nonsteady shear rigid viscoplastic flow in a halfplane initially at rest. Beginning with the initial time moment, the constant tangent stress exceeding a yield stress is given on the boundary. The diffusionvortex solution holds true inside an extending layer with an a priori unknown boundary. The remaining halfplane is immovable in this case. A twodimensional picture of disturbances is imposed on the obtained flow; the picture may then evolve over time. The upper estimates of velocity disturbances by the integral measure of the space $H_2$ are constructed. It is shown that, in a certain range of parameters, the estimating function may decrease up to some point of minimum and only then increase exponentially. The fact of its initial decrease is interpreted as a stabilization of the main flow on a finite time interval.

Privalova V., Prosviryakov E. Y.
Abstract
This paper presents an exact solution to the Oberbeck – Boussinesq system which describes the flow of a viscous incompressible fluid in a plane channel heated by a linear point source. The exact solutions obtained generalize the isothermal Couette flow and the convective motions of Birikh – Ostroumov. A characteristic feature of the proposed class of exact solutions is that they integrate the horizontal gradient of the hydrodynamic fields. An analysis of the solutions obtained is presented and thus a criterion is obtained which explains the existence of countercurrents moving in a nonisothermal viscous incompressible fluid.

Deryabina M. S., Martynov S. I.
Abstract
A procedure is proposed for constructing an approximate periodic solution to the equations of motion of a viscous fluid in an unbounded region in the class of piecewise smooth functions for a given gradient of pressure and temperature for small Reynolds numbers. The procedure is based on splitting the region of the liquid into cells, and finding a solution with boundary conditions corresponding to the periodic function. The cases of two and threedimensional flows of a viscous fluid are considered. It is shown that the solution obtained can be regarded as a flow through a periodic system of point particles placed in the cell corners. It is found that, in a periodic flow, the fluid flow rate per unit of crosssectional area is less than that in a similar Poiseuille flow.

Vetchanin E. V., Gladkov E. S.
Abstract
This paper is concerned with the motion of heavy toroidal bodies in a fluid. For experimental purposes, models of solid tori with a width of 3 cm and external diameters of 10 cm, 12 cm and 15 cm have been fabricated by the method of casting chemically solidifying polyurethane (density 1100 kg/m^{3}). Tracking of the models is performed using the underwater Motion Capture system. This system includes 4 cameras, computer and specialized software. A theoretical description of the motion is given using equations incorporating the influence of inertial forces, friction and circulating motion of a fluid through the hole. Values of the model parameters are selected by means of genetic algorithms to ensure an optimal agreement between experimental and theoretical data.

Gorr G. V., Shchetinina E. K.
Abstract
Two particular cases of the Kovalevskaya solution are studied. A modified Poinsot method is applied for the kinematic interpretation of the body motion. According to this method, the body motion is represented by rolling without sliding of the mobile hodograph of the vector collinear to the angular velocity vector along the stationary hodograph of this vector. Two variants are considered: the first variant is characterized by a plane hodograph of the auxiliary vector; the second variant corresponds to the case where the hodograph of this vector is located on the inertia ellipsoid of the body.

Zubelevich O. E., Salnikova T. V.
Abstract
This article is an extended version of Hadamard’s note devoted to some subtle question that has arisen in the Lagrange top theory. As a rule, this question is not discussed in textbooks.

Kugushev E. I., Popova T. V.
Abstract
The problem of the motion of a homogeneous right circular cylinder with an annular base (a puck) on a horizontal plane with viscous friction is considered. Each point of the base of the puck in contact with the plane is acted upon by the viscous friction force which is proportional to the velocity of this point, and the proportionality coefficient linearly depends on the density of the normal reaction at this point. The density of the normal reaction is determined within the framework of a dynamically consistent model. Some properties of the motion are investigated. In particular, it is shown that for a given direction of the initial angular velocity of the puck, the trajectory of the center of mass of the puck can deviate both to the left and to the right of the straight line directed along the vector of the initial velocity of the center of mass depending on the parameters of the viscous friction model.
