Vol. 17, no. 1
Vol. 17, no. 1, 2021
Kuptsov P. V., Kuptsova A. V., Stankevich N. V.
Abstract
We suggest a universal map capable of recovering the behavior of a wide range of dynamical
systems given by ODEs. The map is built as an artificial neural network whose weights encode
a modeled system. We assume that ODEs are known and prepare training datasets using the
equations directly without computing numerical time series. Parameter variations are taken into
account in the course of training so that the network model captures bifurcation scenarios of the
modeled system. The theoretical benefit from this approach is that the universal model admits
applying common mathematical methods without needing to develop a unique theory for each
particular dynamical equations. From the practical point of view the developed method can be
considered as an alternative numerical method for solving dynamical ODEs suitable for running
on contemporary neural network specific hardware. We consider the Lorenz system, the Rцssler
system and also the Hindmarch – Rose model. For these three examples the network model
is created and its dynamics is compared with ordinary numerical solutions. A high similarity
is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov
exponents.

Pochinka O. V., Nozdrinova E. V.
Abstract
The problem of the existence of an arc with at most countable (finite) number of bifurcations
connecting structurally stable systems (Morse – Smale systems) on manifolds was included in the
list of fifty Palis – Pugh problems at number 33. In 1976 S. Newhouse, J.Palis, F.Takens introduced the concept of a stable arc connecting two structurally stable systems on a manifold. Such an arc does not change its quality properties with small changes. In the same year, S.Newhouse and M.Peixoto proved the existence of a simple arc (containing only elementary bifurcations) between any two Morse – Smale flows. From the result of the work of J. Fliteas it follows that the simple arc constructed by Newhouse and Peixoto can always be replaced by a stable one. For Morse – Smale diffeomorphisms defined on manifolds of any dimension, there are examples of systems that cannot be connected by a stable arc. In this connection, the question naturally arises of finding an invariant that uniquely determines the equivalence class of a Morse – Smale diffeomorphism with respect to the relation of connection by a stable arc (a component of a stable isotopic connection). In the article, the components of the stable isotopic connection of polar gradientlike diffeomorphisms on a twodimensional torus are found under the assumption that all nonwandering points are fixed and have a positive orientation type. 
Mikishanina E. A.
Abstract
This article is devoted to the study of the dynamics of movement of an articulated ntrailer
wheeled vehicle with a controlled leading car. Each link of the vehicle can rotate relative to its
point of fixation. It is shown that, in the case of a controlled leading car, only nonholonomic
constraint equations are sufficient to describe the dynamics of the system, which in turn form
a closed system of differential equations. For a detailed analysis of the dynamics of the system,
the cases of movement of a wheeled vehicle consisting of three symmetric links are considered,
and the leading link (leading car) moves both uniformly along a circle and with a modulo variable
velocity along a certain curved trajectory. The angular velocity remains constant in both cases.
In the first case, the system is integrable and analytical solutions are obtained. In the second
case, when the linear velocity is a periodic function, the solutions of the problem are also periodic.
In numerical experiments with a large number of trailers, similar dynamics are observed.

Polyukhin A. S.
Abstract
Numerical modelling of the thermodynamic properties of plasma mixture is performed using
the Thomas – Fermi model with two different approaches. For this purpose, a numerical
algorithm, as well as program realization, is developed to solve the Thomas – Fermi equations
with quantumexchange corrections. For the first time a comparison between different methods
for taking account of the heterogeneous composition of plasma is made and an algorithm for
estimating the corrections for mixtures is developed.

AlvarezRamirez M., García A., Vidarte J.
Abstract
This article deals with the autonomous twodegreeoffreedom Hamiltonian system with
Armbruster – Guckenheimer – Kim galactic potential in 1:1 resonance depending on two parameters.
We detect periodic solutions and KAM 2tori arising from linearly stable periodic solutions
not found in earlier papers. These are established by using reduction, normalization, averaging
and KAM techniques.

Kholostova O. V.
Abstract
This paper examines the motion of a timeperiodic Hamiltonian system with two degrees
of freedom in a neighborhood of trivial equilibrium. It is assumed that the system depends
on three parameters, one of which is small; when it has zero value, the system is autonomous.
Consideration is given to a set of values of the other two parameters for which, in the autonomous
case, two frequencies of small oscillations of the linearized equations of perturbed motion are
identical and are integer or halfinteger numbers (the case of multiple parametric resonance).
It is assumed that the normal form of the quadratic part of the Hamiltonian does not reduce to
the sum of squares, i.e., the trivial equilibrium of the system is linearly unstable. Using a number
of canonical transformations, the perturbed Hamiltonian of the system is reduced to normal form
in terms through degree four in perturbations and up to various degrees in a small parameter
(systems of first, second and third approximations). The structure of the regions of stability and
instability of trivial equilibrium is investigated, and solutions are obtained to the problems of
the existence, number, as well as (linear and nonlinear) stability of the system’s periodic motions
analytic in fractional or integer powers of the small parameter. For some cases, conditionally
periodic motions of the system are described. As an application, resonant periodic motions of
a dynamically symmetric satellite modeled by a rigid body are constructed in a neighborhood
of its steady rotation (cylindrical precession) on a weakly elliptic orbit and the problem of their
stability is solved.

Bogatenko T. R., Bukh A. V., Strelkova G. I.
Abstract
This paper considers the effects of forced and mutual synchronization of complex spatiotemporal
structures in a twolayer network of nonlocally coupled logistic maps in the presence of
inhomogeneous interlayer coupling. Two different types of coupling topology are considered: the
first one is the sparse interlayer coupling with randomly distributed coupling defects, and the
second type is the cluster interlayer coupling, providing the coupling via designated finite groups
of elements. The latter type of coupling topology is considered for the first time. As a quantitative
measure of the synchronization effect on the network, variance averaged over time and variance
averaged both over time and network elements are used. We analyze how the synchronization
measure changes depending on a degree of the interlayer coupling sparseness. We also identify
a cluster of network elements which can provide almost complete synchronization in the network
under study when the interlayer coupling is introduced along them. This paper is dedicated to the memory of our teacher and scientific supervisor Prof. Vadim S. Anishchenko who passed away last November. 
Raeder T., Tenenev V. A., Koroleva M. R., Mishchenkova O. V.
Abstract
The paper presents a modification of the digital method by S. K. Godunov for calculating
real gas flows under conditions close to a critical state. The method is generalized to the case of
the Van der Waals equation of state using the local approximation algorithm. Test calculations
of flows in a shock tube have shown the validity of this approach for the mathematical description
of gasdynamic processes in real gases with shock waves and contact discontinuity both in areas
with classical and nonclassical behavior patterns. The modified digital scheme by Godunov with
local approximation of the Van der Waals equation by a twoterm equation of state was used for
simulating a spatial flow of real gas based on Navier – Stokes equations in the area of a complex
shape, which is characteristic of the internal space of a safety valve. We have demonstrated that,
under nearcritical conditions, areas of nonclassical gas behavior may appear, which affects the
nature of flows. We have studied nonlinear processes in a safety valve arising from the movement
of the shutoff element, which are also determined by the device design features and the gas
flow conditions.
