 COEDITORSINCHIEF
 Honorary Editor
 Editorial board

Vol. 14, no. 4
Bukh A. V., Strelkova G. I., Anishchenko V. S.
Abstract
Effects of synchronization of chimera states are studied numerically in a twolayer network of nonlocally coupled nonlinear chaotic discretetime 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 crosscorrelation 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
selfsustained oscillations.

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.

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 (quasitwodimensional) 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 selforganization 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.

Mamaev I. S., Tenenev V. A., Vetchanin E. V.
Abstract
This paper addresses the problem of planeparallel 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.

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 Nlinear 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.

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.

Bardin B. S., Panev A. S.
Abstract
We consider a vibrationdriven 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. 
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 gradientlike diffeomorphisms
on a surface or a threedimensional sphere were found. This article is the next step
in this direction. Namely, the author has established that all orientationreversing diffeomorphisms
of a circle are in one component of a simple connection, whereas the simple isotopy class
of an orientationpreserving transformation of a circle is completely determined by the Poincar´e
rotation number.

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.

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 threedimensional 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.

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 leftinvariant subFinsler $\ell_\infty$ norm. We adopt the viewpoint of timeoptimal control theory. By Pontryagin maximum principle, all subFinsler length minimizers belong to one of the following types: abnormal, bangbang, singular, and
mixed. Bangbang controls are piecewise controls with values in the vertices of the set of control parameter. In a previous work, it was shown that bangbang 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 bangbang trajectories, and obtain detailed upper bounds on the number of switchings of optimal control. For bangbang trajectories with low values of the energy integral, we show optimality for arbitrarily large times. The bangbang 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 signdefiniteness 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 bangbang controls may have not more than 11 switchings. For particular patterns of bangbang 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 bangbang trajectories, i.e., the time when these trajectories lose their optimality. This question will be considered in subsequent works. 