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Of all publications in the section: 8
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Article
Kuptsov P., Kuptsova A. V., Stankevich N. Russian Journal of Nonlinear Dynamics. 2021. Vol. 17. No. 1. P. 5-21.

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.

Article
Burov A. A., Nikonov V. Russian Journal of Nonlinear Dynamics. 2020. Vol. 16. No. 2. P. 259-273.

As is well known, many small celestial bodies are of a rather complex shape. Therefore, the study of the dynamics of a spacecraft in their vicinity, based on terms up to the second order of smallness in the expansion of the potential of attraction, seems to be insufficient for an adequate description of the observed dynamical effects related, for example, to positioning of the libration points.In this paper, such effects are demonstrated for spacecraft dynamics in the vicinity of the asteroid (2063) Bacchus. The libration points are computed for various approximations of the gravitational potential. The results of this computation are compared with similar results obtained before for the so-called Sludsky – Werner – Scheeres potential. The dependence of the structure of the regions of possible motions on approximation of the gravitational potential is also studied.

Article
Kuryzhov E., Karatetskaia E., Mints D. Russian Journal of Nonlinear Dynamics. 2021. Vol. 17. No. 2. P. 165-174.

We consider the system of two coupled one-dimensional parabola maps. It is well known that the parabola map is the simplest map that can exhibit chaotic dynamics, chaos in this map appears through an infinite cascade of period-doubling bifurcations. For two coupled parabola maps we focus on studying attractors of two types: those which resemble the well-known discrete Lorenz-like attractors and those which are similar to the discrete Shilnikov attractors. We describe and illustrate the scenarios of occurrence of chaotic attractors of both types.

Article
Barinova M., Гогулина Е. Ю., Pochinka O. Russian Journal of Nonlinear Dynamics. 2021. Vol. 17. No. 3. P. 321-334.

The present paper gives a partial answer to Smale’s question which diagrams can correspond to (A,B)-diffeomorphisms. Model diffeomorphisms of the two-dimensional torus derived by “Smale surgery” are considered, and necessary and sufficient conditions for their topological conjugacy are found. Also, a class G of (A,B)-diffeomorphisms on surfaces which are the connected sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class G realize any connected Hasse diagrams (abstract Smale graph). Examples of diffeomorphisms from G with isomorphic labeled Smale diagrams which are not ambiently Ω-conjugated are constructed. Moreover, a subset G∗ ⊂ G of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient Ω-conjugacy is singled out.

Article
Medvedev T. V., Nozdrinova E., Pochinka O. et al. Russian Journal of Nonlinear Dynamics. 2019. Vol. 15. No. 2. P. 199-211.

We consider the class $G$ of gradient-like orientation preserving diffeomorphisms of the 2-sphere with saddles of negative orientation type. We show that the for every diffeomorphism $f\in G$ every saddle point is fixed. We show that there are exactly three equivalence classes (up to topological conjugacy) $G=G_1\cup G_2\cup G_3$ where a diffeomorphism $f_1\in G_1$ has exactly one saddle and three nodes (one fixed source and two periodic sinks); a diffeomorphism $f_2\in G_2$ has exactly two saddles and four nodes (two periodic sources and two periodic sinks) and a diffeomorphism $f_3\in G_3$ is topologically conjugate to a diffeomorphism $f_1^{-1}$. The main result is the proof that every diffeomorphism $f\in G$ can be connected to the source-sink'' diffeomorphism by a stable arc and this arc contains at most finitely many points of period-doubling bifurcations.

Article
Nozdrinova E. Russian Journal of Nonlinear Dynamics. 2018. Vol. 14. No. 4. P. 543-551.

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, 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 rotation number of Poincare.