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# Vol. 14, no. 4, 2018

 Bukh A. V.,  Strelkova G. I.,  Anishchenko V. S. Abstract Effects of synchronization of chimera states are studied numerically in a two-layer network of nonlocally coupled nonlinear chaotic discrete-time systems. Each layer represents a ring of nonlocally coupled logistic maps in the chaotic mode. A control parameter mismatch is introduced to realize distinct spatiotemporal structures in isolated ensembles. We consider external synchronization of chimeras for unidirectional intercoupling and mutual synchronization in the case of bidirectional intercoupling. Synchronization is quantified by calculating the crosscorrelation coefficient between the symmetric elements of the interacting networks. The same quantity is used to determine finite regions of synchronization inside which the cross-correlation coefficient is equal to 1. The identity of synchronous structures and the existence of finite synchronization regions are necessary and sufficient conditions for establishing the synchronization effect. It is also shown that our results are qualitatively similar to the synchronization of periodic self-sustained oscillations. Keywords: multilayer networks, nonlocal coupling, chimera states, synchronization Citation: Bukh A. V.,  Strelkova G. I.,  Anishchenko V. S., Synchronization of Chimera States in Coupled Networks of Nonlinear Chaotic Oscillators, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 419-433 DOI:10.20537/nd180401
 Doroshenko V. M.,  Kruglov V. P.,  Kuznetsov S. P. Abstract The principle of constructing a new class of systems demonstrating hyperbolic chaotic attractors is proposed. It is based on using subsystems, the transfer of oscillatory excitation between which is provided resonantly due to the difference in the frequencies of small and large (relaxation) oscillations by an integer number of times, accompanied by phase transformation according to an expanding circle map. As an example, we consider a system where a Smale – Williams attractor is realized, which is based on two coupled Bonhoeffer – van der Pol oscillators. Due to the applied modulation of parameter controlling the Andronov – Hopf bifurcation, the oscillators manifest activity and suppression turn by turn. With appropriate selection of the modulation form, relaxation oscillations occur at the end of each activity stage, the fundamental frequency of which is by an integer factor $M = 2, 3, 4, \ldots$ smaller than that of small oscillations. When the partner oscillator enters the activity stage, the oscillations start being stimulated by the $M$-th harmonic of the relaxation oscillations, so that the transformation of the oscillation phase during the modulation period corresponds to the $M$-fold expanding circle map. In the state space of the Poincaré map this corresponds to an attractor of Smale – Williams type, constructed with $M$-fold increase in the number of turns of the winding at each step of the mapping. The results of numerical studies confirming the occurrence of the hyperbolic attractors in certain parameter domains are presented, including the waveforms of the oscillations, portraits of attractors, diagrams illustrating the phase transformation according to the expanding circle map, Lyapunov exponents, and charts of dynamic regimes in parameter planes. The hyperbolic nature of the attractors is verified by numerical calculations that confirm the absence of tangencies of stable and unstable manifolds for trajectories on the attractor (“criterion of angles”). An electronic circuit is proposed that implements this principle of obtaining the hyperbolic chaos and its functioning is demonstrated using the software package Multisim. Keywords: uniformly hyperbolic attractor, Smale – Williams solenoids, Bernoulli mapping, Lyapunov exponents, Bonhoeffer – van der Pol oscillators Citation: Doroshenko V. M.,  Kruglov V. P.,  Kuznetsov S. P., Smale – Williams Solenoids in a System of Coupled Bonhoeffer – van der Pol Oscillators, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 435-451 DOI:10.20537/nd180402
 Maglevanny I. I.,  Smolar V. A.,  Karyakina T. I. Abstract In this paper, we consider the activation processes in a nonlinear bistable system based on a lateral (quasi-two-dimensional) superlattice and study the dynamics of such a system externally driven by a harmonic force. The internal control parameters are the longitudinal applied electric field and the sample temperature. The spontaneous transverse electric field is considered as an order parameter. The forced violations of the order parameter are considered as a response of a system to periodic driving. We investigate the cooperative effects of self-organization and harmonic forcing from the viewpoint of catastrophe theory. Complex nonlinear behaviors including the energetic activation barrier or, more generally, a form of threshold leading to forced bifurcations of dc components of output response accompanied by enhancement of its odd harmonic components were discovered in limited narrow ranges of the control parameters space through numerical simulations. We observed the resonant behaviors of spectral amplification coefficient which is maximized when control parameters are tuned near the critical values of synergetic potential. Keywords: lateral superlattices, applied electric field, spontaneous transverse electric field, nonequilibrium phase transitions, synergetic potential, forced bifurcations, resonant enhancement Citation: Maglevanny I. I.,  Smolar V. A.,  Karyakina T. I., Weak Signals Amplification Through Controlled Bifurcations in Quasi-Two-Dimensional Electron Gas, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 453-472 DOI:10.20537/nd180403
 Mamaev I. S.,  Tenenev V. A.,  Vetchanin  E. V. Abstract This paper addresses the problem of plane-parallel motion of the Zhukovskii foil in a viscous fluid. Various motion regimes of the foil are simulated on the basis of a joint numerical solution of the equations of body motion and the Navier – Stokes equations. According to the results of simulation of longitudinal, transverse and rotational motions, the average drag coefficients and added masses are calculated. The values of added masses agree with the results published previously and obtained within the framework of the model of an ideal fluid. It is shown that between the value of circulation determined from numerical experiments, and that determined according to the model of and ideal fluid, there is a correlation with the coefficient $\mathcal{R} = 0.722$. Approximations for the lift force and the moment of the lift force are constructed depending on the translational and angular velocity of motion of the foil. The equations of motion of the Zhukovskii foil in a viscous fluid are written taking into account the found approximations and the drag coefficients. The calculation results based on the proposed mathematical model are in qualitative agreement with the results of joint numerical solution of the equations of body motion and the Navier – Stokes equations. Keywords: Zhukovskii foil, Navier – Stokes equations, joint solution of equations, finitedimensional model, viscous fluid, circulation, sharp edge Citation: Mamaev I. S.,  Tenenev V. A.,  Vetchanin  E. V., Dynamics of a Body with a Sharp Edge in a Viscous Fluid, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 473-494 DOI:10.20537/nd180404
 Fimin N. N.,  Chechetkin V. M. Abstract Geometrization of the description of vortex hydrodynamic systems can be made on the basis of the introduction of the Monge – Clebsch potentials, which leads to the Hamiltonian form of the original Euler equations. For this, we construct the kinetic Lagrange potential with the help of the flow velocity field, which is preliminarily determined through a set of scalar Monge potentials, and thermodynamic relations. The next step is to transform the resulting Lagrangian by means of the Legendre transformation to the Hamiltonian function and correctly introduce the generalized impulses canonically conjugate to the configuration variables in the new phase space of the dynamical system. Next, using the Hamiltonian function obtained, we define the Hamiltonian space on the cotangent bundle over the Monge potential manifold. Calculating the Hessian of the Hamiltonian, we obtain the coefficients of the fundamental tensor of the Hamiltonian space defining its metric. Next, we determine analogs of the Christoffel coefficients for the N-linear connection. Considering the Euler – Lagrange equations with the connectivity coefficients obtained, we arrive at the geodesic equations in the form of horizontal and vertical paths in the Hamiltonian space. In the case under study, nontrivial solutions can have only differential equations for vertical paths. Analyzing the resulting system of equations of geodesic motion from the point of view of the stability of solutions, it is possible to obtain important physical conclusions regarding the initial hydrodynamic system. To do this, we investigate a possible increase or decrease in the infinitesimal distance between the geodesic vertical paths (solutions of the corresponding system of Jacobi – Cartan equations). As a result, we can formulate very general criterions for the decay and collapse of a vortex continual system. Keywords: vortex dynamics, geodesic deviation, Monge manifold, metric tensor Citation: Fimin N. N.,  Chechetkin V. M., The Possibility of Introducing of Metric Structure in Vortex Hydrodynamic Systems, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 495-501 DOI:10.20537/nd180405
 Markeev A. P. Abstract The motion of a dynamically symmetric rigid body relative to its center of mass in the central Newtonian gravitational field in a circular orbit is investigated. This problem involves motion (called conical precession) where the dynamical symmetry axis of the body is located all the time in the plane perpendicular to the velocity vector of the center of mass of the body and makes a constant angle with the direction of the radius vector of the center of mass relative to the attracting center. This paper deals with a special case in which this angle is $\pi/4$ and the ratio between the polar and the equatorial principal central moments of inertia of the body is equal to the number $2/3$ or is close to it. In this case, the conical precession is stable with respect to the angles that define the position of the symmetry axis in an orbital coordinate system and with respect to the time derivatives of these angles, and the frequencies of small (linear) oscillations of the symmetry axis are equal or close to each other (that is, the 1:1 resonance takes place). Using classical perturbation theory and modern numerical and analytical methods of nonlinear dynamics, a solution is presented to the problem of the existence, bifurcations and stability of periodic motions of the symmetry axis of a body which are generated from its relative (in the orbital coordinate system) equilibrium corresponding to conical precession. The problem of the existence of conditionally periodic motions is also considered. Keywords: resonance, stability, oscillations, canonical transformations Citation: Markeev A. P., On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 503-518 DOI:10.20537/nd180406
 Bardin B. S.,  Panev A. S. Abstract We consider a vibration-driven system which consists of a rigid body and an internal mass. The internal mass is a particle moving in a circle inside the body. The center of the circle is located at the mass center of the body and the absolute value of particle velocity is a constant. The body performs rectilinear motion on a horizontal plane, whereas the particle moves in a vertical plane. We suppose that dry friction acts between the plane and the body. We have investigated the dynamics of the above system in detail and given a full description of the body’s motion for any values of its initial velocity. In particular, it is shown that there always exists a periodic mode of motion. Depending on parameter values, one of three types of this periodic mode takes place. At any initial velocity the body either enters a periodic mode during a finite time interval or it asymptotically approaches the periodic mode. Keywords: periodic motion, dry friction, rigid body, vibration-driven system Citation: Bardin B. S.,  Panev A. S., On the Motion of a Body with a Moving Internal Mass on a Rough Horizontal Plane, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 519-542 DOI:10.20537/nd180407
 Nozdrinova E. V. Abstract The problem of the existence of a simple arc connecting two structurally stable systems on a closed manifold is included in the list of the fifty most important problems of dynamical systems. This problem was solved by S. Newhouse and M. Peixoto for Morse – Smale flows on an arbitrary closed manifold in 1980. As follows from the works of Sh. Matsumoto, P. Blanchard, V. Grines, E.Nozdrinova, and O.Pochinka, for the Morse – Smale cascades, obstructions to the existence of such an arc exist on closed manifolds of any dimension. In these works, necessary and sufficient conditions for belonging to the same simple isotopic class for gradient-like diffeomorphisms on a surface or a three-dimensional sphere were found. This article is the next step in this direction. Namely, the author has established that all orientation-reversing diffeomorphisms of a circle are in one component of a simple connection, whereas the simple isotopy class of an orientation-preserving transformation of a circle is completely determined by the Poincar´e rotation number. Keywords: rotation number, simple arc Citation: Nozdrinova E. V., Rotation Number as a Complete Topological Invariant of a Simple Isotopic Class of Rough Transformations of a Circle, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 543-551 DOI:10.20537/nd180408
 Dzhalilov A.,  Mayer D.,  Djalilov S.,  Aliyev A. Abstract M. Herman showed that the invariant measure $\mu_h$ of a piecewise linear (PL) circle homeomorphism $h$ with two break points and an irrational rotation number $\rho_{h}$ is absolutely continuous iff the two break points belong to the same orbit. We extend Herman's result to the class P of piecewise $C^{2+\varepsilon}$-circle maps $f$ with an irrational rotation number $\rho_f$ and two break points $a_{0}, c_{0}$, which do not lie on the same orbit and whose total jump ratio is $\sigma_f=1$, as follows: if $\mu_f$ denotes the invariant measure of the $P$-homeomorphism $f$, then for Lebesgue almost all values of $\mu_f([a_0, c_{0}])$ the measure $\mu_f$ is singular with respect to Lebesgue measure. Keywords: piecewise-smooth circle homeomorphism, break point, rotation number, invariant measure Citation: Dzhalilov A.,  Mayer D.,  Djalilov S.,  Aliyev A., An Extention of Herman’s Theorem for Nonlinear Circle Maps with Two Breaks, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 553-577 DOI:10.20537/nd180409
 Podobryaev A. Abstract We give an example of a Riemannian manifold homeomorphic to a sphere such that its diameter cannot be realized as a distance between antipodal points. We consider a Berger sphere, i.e., a three-dimensional sphere with Riemannian metric that is compressed along the fibers of the Hopf fibration. We give a condition for a Berger sphere to have the desired property. We use our previous results on a cut locus of Berger spheres obtained by the method from geometric control theory. Keywords: diameter, $SU_2$, Berger sphere, antipodal points, cut locus, geometric control theory Citation: Podobryaev A., Antipodal Points and Diameter of a Sphere, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 579-581 DOI:10.20537/nd180410
 Sachkov Y. L. Abstract The Cartan group is the free nilpotent Lie group of step 3, with 2 generators. This paper studies the Cartan group endowed with the left-invariant sub-Finsler $\ell_\infty$ norm. We adopt the viewpoint of time-optimal control theory. By Pontryagin maximum principle, all sub-Finsler length minimizers belong to one of the following types: abnormal, bang-bang, singular, and mixed. Bang-bang controls are piecewise controls with values in the vertices of the set of control parameter. In a previous work, it was shown that bang-bang trajectories have a finite number of patterns determined by values of the Casimir functions on the dual of the Cartan algebra. In this paper we consider, case by case, all patterns of bang-bang trajectories, and obtain detailed upper bounds on the number of switchings of optimal control. For bang-bang trajectories with low values of the energy integral, we show optimality for arbitrarily large times. The bang-bang trajectories with high values of the energy integral are studied via a second order necessary optimality condition due to A. Agrachev and R.Gamkrelidze. This optimality condition provides a quadratic form, whose sign-definiteness is related to optimality of bangbang trajectories. For each pattern of these trajectories, we compute the maximum number of switchings of optimal control. We show that optimal bang-bang controls may have not more than 11 switchings. For particular patterns of bang-bang controls, we obtain better bounds. In such a way we improve the bounds obtained in previous works. On the basis of results of this work we can start to study the cut time along bang-bang trajectories, i.e., the time when these trajectories lose their optimality. This question will be considered in subsequent works. Keywords: sub-Finsler geometry, optimal control, switchings, bang-bang trajectories Citation: Sachkov Y. L., Optimal Bang-Bang Trajectories in Sub-Finsler Problem on the Cartan Group, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 583-593 DOI:10.20537/nd180411
 Abstract Citation: Vyacheslav Vladimirovich Meleshko, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 595-600 DOI:10.20537/nd180412

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