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Kilin A. A., Ivanova T. B.
Abstract
This paper addresses the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that there is no slipping of the sphere as it rolls in the direction of the projection of the symmetry axis onto the supporting plane. It is also assumed that, in the direction perpendicular to the abovementioned one, the sphere can slip relative to the plane. Examples of realization of the abovementioned nonholonomic constraint are given. Equations of motion are obtained and their first integrals are found. It is shown that the system under consideration admits a redundant set of first integrals, which makes it possible to perform reduction to a system with one degree of freedom.

Garashchuk I. R., Sinelshchikov D. I.
Abstract
We study a model of three Hindmarsh – Rose neurons with directional electrical connections.
We consider two fullyconnected neurons that form a slave group which receives the signal from
the master neuron via a directional coupling. We control the excitability of the neurons by setting
the constant external currents. We study the possibility of excitation of the slave system in the
stable resting state by the signal coming from the master neuron, to make it fire spikes/bursts
tonically. We vary the coupling strength between the master and the slave systems as another
control parameter. We calculate the borderlines of excitation by different types of signal in the
control parameter space. We establish which of the resulting dynamical regimes are chaotic.
We also demonstrate the possibility of excitation by a single burst or a spike in areas of control
parameters, where the slave system is bistable. We calculate the borderlines of excitation by
a single period of the excitatory signal.

Tsirlin A. M.
Methods of Simplifying Optimal Control Problems, Heat Exchange and Parametric Control of Oscillators
Abstract
Methods of simplifying optimal control problems by decreasing the dimension of the space of
states are considered. For this purpose, transition to new phase coordinates or conversion of the
phase coordinates to the class of controls is used. The problems of heat exchange and parametric
control of oscillators are given as examples: braking/swinging of a pendulum by changing the
length of suspension and variation of the energy of molecules’ oscillations in the crystal lattice by
changing the state of the medium (exposure to laser radiation). The last problem corresponds
to changes in the temperature of the crystal.

Udalov A. A., Uleysky M. Y., Budyansky M. V.
Abstract
A dynamically consistent model of a meandering jet stream with two Rossby waves obtained
using the law of conservation of potential vorticity is investigated. Stationary points are found in
the phase space of advection equations and the type of their stability is determined analytically.
All topologically different flow regimes and their bifurcations are found for the stationary model
(taking into account only the first Rossby wave). The results can be used in the study of
Lagrangian transport, mixing, and chaotic advection in problems of crossfrontal transport in
geophysical flows with meandering jets.

Indeitsev D. A., Zavorotneva E. V., Lukin A. V., Popov I. A., Igumnova V. S.
Abstract
This article presents an analytical study of the dynamics of a micromechanical integrating
gyroscope with a disk resonator. A discrete dynamic model of the resonator is obtained, taking
into account the axial anisotropy of its mass and stiffness properties, as well as the action of the
electrical control system of oscillations. An analysis of the spectral problem of disk vibrations
in the plane is carried out. The nonlinear dynamics of the resonator in the regimes of free
and parametrically excited vibrations are investigated. In the mode of parametric oscillations,
qualitative dependencies of the gyroscopic drift on the operating voltage, angular velocity and
parameters of defects are obtained.

Baranov D. A., Grines V. Z., Pochinka O. V., Chilina E. E.
Abstract
In this paper, following J. Nielsen, we introduce a complete characteristic of orientationpreserving
periodic maps on the twodimensional torus. All admissible complete characteristics
were found and realized. In particular, each of the classes of orientationpreserving periodic
homeomorphisms on the 2torus that are nonhomotopic to the identity is realized by an algebraic
automorphism. Moreover, it is shown that the number of such classes is finite. According to
V. Z. Grines and A.Bezdenezhnykh, any gradientlike orientationpreserving diffeomorphism of
an orientable surface is represented as a superposition of the time1 map of a gradientlike flow
and some periodic homeomorphism. Thus, the results of this work are directly related to the
complete topological classification of gradientlike diffeomorphisms on surfaces.

Golokolenov A. V., Savin D. V.
Abstract
This article deals with the dynamics of a pulsedriven selfoscillating system — the Van
der Pol oscillator — with the pulse amplitude depending on the oscillator coordinate. In the
conservative limit the “stochastic web” can be obtained in the phase space when the function
defining this dependence is a harmonic one. The paper focuses on the case where the frequency of
external pulses is four times greater than the frequency of the autonomous system. The results of
a numerical study of the structure of both parameter and phase planes are presented for systems
with different forms of external pulses: the harmonic amplitude function and its power series
expansions. Complication of the pulse amplitude function results in the complication of the
parameter plane structure, while typical scenarios of transition to chaos visible in the parameter
plane remain the same in different cases. In all cases the structure of bifurcation lines near the
border of chaos is typical of the existence of the Hamiltonian type critical point. Changes in
the number and the relative position of coexisting attractors are investigated while the system
approaches the conservative limit. A typical scenario of destruction of attractors with a decrease
in nonlinear dissipation is revealed, and it is shown to be in good agreement with the theory
of 1:4 resonance. The number of attractors of period 4 seems to grow infinitely with the decrease
of dissipation when the pulse amplitude function is harmonic, while in other cases all attractors
undergo destruction at certain values of dissipation parameters after the birth of highperiod
periodic attractors.

Popkov Y. S.
Abstract
This paper considers dynamic systems with an entropy operator described by a perturbed
constrained optimization problem. Oscillatory processes are studied for periodic systems with
the following property: the entire system has the same period as the process generated by its
linear part. Existence and uniqueness conditions are established for such oscillatory processes,
and a method is developed to determine their form and parameters. Also, the general case of
noncoincident periods is analyzed, and a method is proposed to determine the form, parameters,
and the period of such oscillations. Almost periodic processes are investigated, and existence
and uniqueness conditions are proved for them as well.

Semernik I. V., Bender O. V., Tarasenko A. A., Samonova C. V.
Abstract
In this article, a method for increasing the noise immunity of an underwater wireless optical
communication system by applying chaotic oscillations is considered. To solve this problem, it
is proposed to use modulation methods based on dynamical chaos at the physical level of the
communication channel. Communication channel modeling is implemented by calculating the impulse response using a numerical solution of the radiation transfer equation by the Monte Carlo method. The following modulation methods based on the correlation processing of the received signal are considered: chaotic mode switching, chaotic onoff keying (COOK). Onoff keying (OOK) modulation was chosen as a test modulation method to assess the degree of noise immunity of the modulation methods under study. An analysis of the noise immunity of an underwater optical communication channel with a change in its length and parameters of the aquatic environment, which affect the absorption and scattering of optical radiation in the communication channel, is carried out. It is shown that modulation methods based on the phenomenon of dynamic chaos and correlation processing can improve the noise immunity of underwater wireless communication systems. This provides the possibility of signal recovery at negative values of the signaltonoise ratio. It is shown that the considered modulation methods (COOK and switching of chaotic modes) in combination with the correlation processing of the signal at the physical level of the communication channel provide an advantage of about 15 dB compared to OOK modulation. 