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# Vol. 15, no. 2, 2019

 Kuznetsov S. P. Abstract It is shown that on the basis of a cellular neural network (CNN) composed, e.g., of six cells, it is possible to design a chaos generator with an attractor being a kind of Smale – Williams solenoid, which provides chaotic dynamics that is rough (structurally stable), as follows from respective fundamental mathematical theory. In the context of the technical device, it implies insensitivity to small variations of parameters, manufacturing imperfections, interferences, etc. Results of numerical simulations and circuit simulation in the Multisim environment are presented. The proposed circuit is the first example of an electronic system where the role of the angular coordinate for the Smale – Williams attractor is played by the spatial phase of the sequence of patterns. It contributes to the collection of feasible systems with hyperbolic attractors and thus promotes filling with real content and promises practical application for the hyperbolic theory, which is an important and deep sector of the modern mathematical theory of dynamical systems. Keywords: cellular neural network, chaos, attractor, pattern, dynamics, structural stability Citation: Kuznetsov S. P., Generation of Robust Hyperbolic Chaos in CNN, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp. 109-124 DOI:10.20537/nd190201
 Lamzoud K.,  Assoudi R.,  Bouisfi F.,  Chaoui M. Abstract Based on the assumption of low Reynolds number, the flow around a spherical particle settling towards a corrugated wall in a fluid at rest is described by Stokes equations. In the case of the small amplitude of wall roughness, the asymptotic expansion coupled with the Lorentz reciprocal theorem are used to derive analytical expressions of the hydrodynamic effects due to wall roughness. The evolution of these effects in terms of roughness parameters and also the sphere-wall distance are discussed. Keywords: Stokes equations, low Reynolds number, roughness effects, asymptotic expansion, Lorentz reciprocal theorem Citation: Lamzoud K.,  Assoudi R.,  Bouisfi F.,  Chaoui M., A Spherical Particle Settling Towards a Corrugated Wall, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp. 125-134 DOI:10.20537/nd190202
 Petrov A. G. Abstract The Joukowski – Chaplygin condition, which allows us to determine the circulation of the flow past contour with a sharp edge, is one of the most important achievements of S. A.Chaplygin, whose 150 birthday is celebrated this year. This research is devoted to this problem. We consider the flow past of a lattice of airfoils by a potential fluid flow. The profile line is determined parametrically by an equation in the form of two dependences of the Cartesian coordinates on the parameter. For a smooth closed loop the Cartesian coordinates are periodic analytical functions. Their Fourier series coefficients decrease exponentially depending on the harmonic number. In the meantime, for a sharp edge loop they decrease much slower — inversely to the square of the harmonic number. Using the symmetric continuation of the profile with a sharp edge, a method of presenting it as a Fourier series with exponentially decreasing coefficients is proposed. Based on this idea, a quickly converging numerical scheme for the computation of the flow past airfoils lattice with a sharp edge by a potential fluid flow has been developed. The problem is reduced to a linear integro-differential equation on the lattice contour, and then, using specially developed quadrature formulas, is approximated by a linear system of equations. The quadrature formulas converge exponentially with respect to the number of points on the profile and can be rather simply expressed analytically. Thanks to its quick convergence and high accuracy, this method allows one to optimize profiles by using a direct method by any given integral characteristics. We can find the distribution of the shear stress and the breakaway point on the calculated velocity distribution on the profile from the solution of the boundary layer equation. In the method suggested we do not need to perform the difficult work of constructing the lattice. Also, the problem of scheme viscosity at high Reynolds numbers is omitted. Keywords: lattice of airfoils, potential fluid flow, sharp edge loop Citation: Petrov A. G., Saturation-Free Numerical Scheme for Computing the Flow Past a Lattice of Airfoils with a Sharp Edge, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp. 135-143 DOI:10.20537/nd190203
 Gouskov A. M.,  Guskov M. A.,  Tung D. D.,  Panovko G. Y. Abstract This work presents nonlinear dynamics modeling results for an investigation of continuous cut stability in multicutter turning. The dynamics modeling of the multicutter turning process is carried out through the complete mathematical model of nonlinear dynamics. The dynamic stability of the system is estimated through the possibility of self-oscillations generation (Poincaré – Andronov –Hopf bifurcation) of the cutters with lobes of the stability diagram. This paper analyzes the relationship of the axial offset and the cutter angular position for compensation of the system parameters. As a result, the analysis of the influence of the technological system parameters on the chip thickness, their cross-sectional shape and the stability of the system is carried out. Keywords: multicutter turning, dynamics, modeling, bifurcation analysis, steady cutting stability conditions Citation: Gouskov A. M.,  Guskov M. A.,  Tung D. D.,  Panovko G. Y., Nonlinear Regenerative Dynamics Analysis of the Multicutter Turning Process, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp. 145-158 DOI:10.20537/nd190204
 Shamin A. Y. Abstract This paper addresses the problem of the motion of the Chaplygin sleigh, a rigid body with three legs in contact with a horizontal plane, one of which is equipped with a semicircular skate orthogonal to the horizontal plane. The problem is considered in a nonholonomic setting: assuming that the blade cannot slide in a direction perpendicular to its plane, but unlike the Chaplygin problem, there is a dry friction force in the skate that is directed along the skate, along which the blade plane and the reference plane intersect. It is also assumed that at the two other points of support there are dry friction forces. The equations of motion of the Chaplygin sleigh are obtained, and a number of properties are proved. It is proved that the movement ceases in finite time. The possibility of realizing the nonnegativity of normal reactions is discussed. The case of static friction is studied when the blade velocity is $v=0$. A region of stagnation where the system rotates about a fixed vertical axis is found. On this set, the equations of motion are integrated and the law of variation of the angular velocity is found. Examples of trajectories of the sleigh are given. A qualitative description of the motion is obtained: the behavior of the phase curves in a neighborhood of the equilibrium point is investigated depending on the geometric and mass characteristics of the system. Keywords: dry friction, Chaplygin sleigh Citation: Shamin A. Y., On the Motion of the Chaplygin Sleigh on a Horizontal Plane with Dry Friction at Three Points of Contact, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp. 159-169 DOI:10.20537/nd190205
 Ivanova T. B. Abstract This paper is concerned with the rolling of a homogeneous ball with slipping on a uniformly rotating horizontal plane. We take into account viscous friction forces arising when there is slipping at the contact point. It is shown that, as the coefficient of viscosity tends to infinity, the solution of the generalized problem on each fixed time interval tends to a solution of the corresponding nonholonomic problem. Keywords: rotating surface, turntable, nonholonomic constraint, rolling ball, sliding, viscous friction Citation: Ivanova T. B., The Rolling of a Homogeneous Ball with Slipping on a Horizontal Rotating Plane, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp. 171-178 DOI:10.20537/nd190206
 Inozemtsev V. I. Abstract A purely algebraic method is proposed for the construction of zonal spherical functions (ZSF) on symmetric spaces $X_{n}^{-} = SL(n,Q)/Sp(n)$ and eigenfunctions of the hyperbolic Sutherland operator connected with them. Examples of the explicit calculations of the coefficients determining the structure of ZSF are given. Keywords: hyperbolic Sutherland operator, permutation group, zonal spherical functions Citation: Inozemtsev V. I., On the Structure of Zonal Spherical Functions on Symmetric Spaces of Negative Curvature of Type AII, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp. 179-186 DOI:10.20537/nd190207
 Morozov A. D.,  Morozov K. E. Abstract We study quasi-periodic nonconservative perturbations of two-dimensional Hamiltonian systems. We suppose that there exists a region $D$ filled with closed phase curves of the unperturbed system and consider the problem of global dynamics in $D$. The investigation includes examining the behavior of solutions both in $D$ (the existence of invariant tori, the finiteness of the set of splittable energy levels) and in a neighborhood of the unperturbed separatrix (splitting of the separatrix manifolds). The conditions for the existence of homoclinic solutions are stated. We illustrate the research with the Duffing – Van der Pole equation as an example. Keywords: resonances, quasi-periodic, periodic, averaged system, phase curves, equilibrium states, limit cycles, separatrix manifolds Citation: Morozov A. D.,  Morozov K. E., Global Dynamics of Systems Close to Hamiltonian Ones Under Nonconservative Quasi-periodic Perturbation, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp. 187-198 DOI:10.20537/nd190208
 Medvedev T. V.,  Nozdrinova E. V.,  Pochinka O. V.,  Shadrina E. V. Abstract We consider the class $G$ of gradient-like orientation-preserving diffeomorphisms of the 2-sphere with saddles of negative orientation type. We show that the for every diffeomorphism $f\in G$ every saddle point is fixed. We show that there are exactly three equivalence classes (up to topological conjugacy) $G=G_1\cup G_2\cup G_3$ where a diffeomorphism $f_1\in G_1$ has exactly one saddle and three nodes (one fixed source and two periodic sinks); a diffeomorphism $f_2\in G_2$ has exactly two saddles and four nodes (two periodic sources and two periodic sinks) and a diffeomorphism $f_3\in G_3$ is topologically conjugate to a diffeomorphism $f_1^{-1}$. The main result is the proof that every diffeomorphism $f\in G$ can be connected to the source-sink'' diffeomorphism by a stable arc and this arc contains at most finitely many points of period-doubling bifurcations. Keywords: sink-source map, stable arc Citation: Medvedev T. V.,  Nozdrinova E. V.,  Pochinka O. V.,  Shadrina E. V., On a Class of Isotopic Connectivity of Gradient-like Maps of the 2-sphere with Saddles of Negative Orientation Type, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp. 199-211 DOI:10.20537/nd190209

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