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Condolence note

Professor Alexey V. Borisov, Editor-in-Chief of the journal RJND, suddenly passed away at the age of 55 on January 24, 2021.
This is a tremendous loss for all of us. We will always remember him as a great scientist and an outstanding personality.

Vol. 17, no. 1

Citation: In memory of Professor Alexey V. Borisov (27.03.1965–24.01.2021), Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp. 3-3
Kuptsov P. V.,  Kuptsova A. V.,  Stankevich N. V.
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.
Keywords: neural network, dynamical system, numerical solution, universal approximation theorem, Lyapunov exponents
Citation: Kuptsov P. V.,  Kuptsova A. V.,  Stankevich N. V., Artificial Neural Network as a Universal Model of Nonlinear Dynamical Systems, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp. 5-21
Pochinka O. V.,  Nozdrinova E. V.
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 gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.
Keywords: stable arc, saddle-node, gradient-like diffeomorphism, two-dimensional torus
Citation: Pochinka O. V.,  Nozdrinova E. V., Stable Arcs Connecting Polar Cascades on a Torus, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp. 23-37
Mikishanina E. A.
This article is devoted to the study of the dynamics of movement of an articulated n-trailer 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.
Keywords: wheeled vehicle, nonholonomic constraint, mathematical model, dynamics of system
Citation: Mikishanina E. A., Dynamics of a Controlled Articulated $n$-trailer Wheeled Vehicle, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp. 39-48
Polyukhin A. S.
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 quantum-exchange 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.
Keywords: computational model, magnetized plasma, numerical method, thermodynamic property
Citation: Polyukhin A. S., Modelling of the Thermodynamic Properties of the Plasma Mixture, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp. 49-57
Alvarez-Ramirez M.,  García A.,  Vidarte J.
This article deals with the autonomous two-degree-of-freedom Hamiltonian system with Armbruster – Guckenheimer – Kim galactic potential in 1:1 resonance depending on two parameters. We detect periodic solutions and KAM 2-tori arising from linearly stable periodic solutions not found in earlier papers. These are established by using reduction, normalization, averaging and KAM techniques.
Keywords: galactic potential, Hamiltonian system, normalization and reduction, KAM tori, reduced space, periodic orbits
Citation: Alvarez-Ramirez M.,  García A.,  Vidarte J., Armbruster – Guckenheimer – Kim Hamiltonian System in 1:1 Resonance, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp. 59-76
Kholostova O. V.
This paper examines the motion of a time-periodic 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 half-integer 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.
Keywords: multiple parametric resonance, method of normal forms, stability, nonlinear oscillations, periodic motions, dynamically symmetric satellite, cylindrical precession
Citation: Kholostova O. V., On Nonlinear Oscillations of a Near-Autonomous Hamiltonian System in the Case of Two Identical Integer or Half-Integer Frequencies, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp. 77-102
Bogatenko T. R.,  Bukh A. V.,  Strelkova G. I.
This paper considers the effects of forced and mutual synchronization of complex spatiotemporal structures in a two-layer 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.
Keywords: synchronization, sparse links, chimera state, multilayer network, interlayer coupling
Citation: Bogatenko T. R.,  Bukh A. V.,  Strelkova G. I., Peculiarities of Synchronization in a Two-Layer Network of Chaotic Maps with Inhomogeneous Interlayer Coupling, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp. 103-117
Raeder T.,  Tenenev V. A.,  Koroleva M. R.,  Mishchenkova O. V.
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 gas-dynamic 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 two-term 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 near-critical 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 shut-off element, which are also determined by the device design features and the gas flow conditions.
Keywords: real gas, Van der Waals equation, critical state of substance, Godunov’s method
Citation: Raeder T.,  Tenenev V. A.,  Koroleva M. R.,  Mishchenkova O. V., Nonlinear Processes in Safety Systems for Substances with Parameters Close to a Critical State, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp. 119-138