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# 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 two-hour 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. Keywords: phase synchronization, autonomic regulation, self-oscillatory circuit, cardiovascular system, data analysis Citation: Skazkina V.,  Kiselev A. R.,  Borovkova  E. I.,  Ponomarenko V. I.,  Prokhorov M. D.,  Karavaev A. S., Estimation of synchronization of contours of vegetative regulation of circulation from long time records, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 3-12 DOI:10.20537/nd1801001
 Kulakov M. P.,  Frisman E. Y. Abstract The article is devoted to the model of spatial-temporal dynamics of age-structured populations coupled by migration. The dynamics of a single population is described by a two-dimensional 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 space-time 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 space-time 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 spatial-temporal 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. Keywords: population, multistability, coupled map lattice, synchronization, clustering, chimera, basin of attraction Citation: Kulakov M. P.,  Frisman E. Y., Clustering and chimeras in the model of the spatial-temporal dynamics of agestructured populations, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 13-31 DOI:10.20537/nd1801002
 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 near-Hamiltonian systems, the program law is constructed for test self-oscillations. Test self-oscillations are to be used for identification of friction coefficients. Bifurcation diagrams are constructed that describe the dependence between amplitudes of self-oscillations 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. Keywords: identification of friction, small parameter, steady motions, controlled dynamic system, bifurcation diagrams Citation: Vasiukova O. E.,  Klimina L. A., Modelling of self-oscillations of a controlled pendulum with respect to a friction torque depending on a normal reaction in a joint, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 33-44 DOI:10.20537/nd1801003
 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 so-called “dog-bone” shape. It turns out that approximation of these bodies by dumb-bells, 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 so-called $K$-mean algorithm proposed by the prominent Polish mathematician H. Steinhaus. Keywords: $K$-means algorithm, small celestial bodies, mesh representation of an asteroid’s surface Citation: Burov A. A.,  Guerman A.,  Raspopova E.,  Nikonov V., On the use of the $K$-means algorithm for determination of mass distributions in dumbbell-like celestial bodies, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 45-52 DOI:10.20537/nd1801004
 Fimin N. N.,  Chechetkin V. M. Abstract The properties of quasi-linear 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 non-Hamiltonian) first-order autonomous differential equations. Examples are given of the use of the developed formalism for systems of gravitating particles in post-Newtonian 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. Keywords: Liouville equation, quasi-linear equations, hydrodynamic substitution, Monge potentials, equations with the same principal part Citation: Fimin N. N.,  Chechetkin V. M., Application of the hydrodynamic substitution for systems of equations with the same principal part, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 53-61 DOI:10.20537/nd1801005
 Georgievsky D. V. Abstract This paper is concerned with obtaining the parameters of a nonsteady shear rigid viscoplastic flow in a half-plane initially at rest. Beginning with the initial time moment, the constant tangent stress exceeding a yield stress is given on the boundary. The diffusion-vortex solution holds true inside an extending layer with an a priori unknown boundary. The remaining half-plane is immovable in this case. A two-dimensional 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. Keywords: viscoplastic solid, rigid domain, yield stress, diffusion, vortex layer, nonsteady shear, disturbance, quadratic functional Citation: Georgievsky D. V., On the diffusion of a rigid viscoplastic vortex layer, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 63-67 DOI:10.20537/nd1801006
 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. Keywords: Couette flow, Birikh – Ostroumova flow, planar Rayleigh – Benard convection, quadratic heating, exact solution, counterflow Citation: Privalova V.,  Prosviryakov E. Y., Steady convective Coutte flow for quadratic heating of the lower boundary fluid layer, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 69-79 DOI:10.20537/nd1801007
 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 three-dimensional 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 cross-sectional area is less than that in a similar Poiseuille flow. Keywords: viscous fluid, periodic solution, piecewise function, gradient, pressure, temperature Citation: Deryabina M. S.,  Martynov S. I., Periodic flow of a viscous fluid with a predetermined pressure and temperature gradient, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 81-97 DOI:10.20537/nd1801008
 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/m3). 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. Keywords: fall through a fluid, torus, body with a hole, multiply connected body, finitedimensional model, object tracking, genetic algorithms Citation: Vetchanin  E. V.,  Gladkov E. S., Identification of parameters of the model of toroidal body motion using experimental data, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 99-121 DOI:10.20537/nd1801009
 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. Keywords: Kovalevskaya’s solution, Poinsot’s method Citation: Gorr G. V.,  Shchetinina E. K., On the motion of a heavy rigid body in two special cases of S.V.Kovalevskaya’s solution, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 123-138 DOI:10.20537/nd1801010
 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. Keywords: gyroscopes, motion of a rigid body with a fixed point Citation: Zubelevich O. E.,  Salnikova T. V., A note on Lagrange’s top theory, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 139-143 DOI:10.20537/nd1801011
 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. Keywords: puck with annular base, viscous friction, coefficient of friction, dynamically consistent model of normal reactions Citation: Kugushev E. I.,  Popova T. V., On the motion of a puck on a horizontal plane in the model of viscous friction with variable coefficient, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp. 145-153 DOI:10.20537/nd1801012

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