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Regular version of the site
Of all publications in the section: 21
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Article
Burov A. A., Guerman A., Nikonov V. Regular and Chaotic Dynamics. 2020. Vol. 25. No. 1. P. 121-130.

Invariant surfaces play a crucial role in the dynamics of mechanical systems separating regions filled with chaotic behavior. Cases where such surfaces can be found are rare enough. Perhaps the most famous of these is the so-called Hess case in the mechanics of a heavy rigid body with a fixed point. We consider here the motion of a non-autonomous mechanical pendulum-like system with one degree of freedom. The conditions of existence for invariant surfaces of such a system corresponding to non-split separatrices are investigated. In the case where an invariant surface exists, combination of regular and chaotic behavior is studied analytically via the Poincaré – Mel'nikov separatrix splitting method, and numerically using the Poincaré maps

Added: Nov 12, 2020
Article
Stankevich N., Dvorak A., Astakhov V. et al. Regular and Chaotic Dynamics. 2018. Vol. 23. No. 1. P. 120-126.

The dynamics of two coupled antiphase driven Toda oscillators is studied. We demonstrate three different routes of transition to chaotic dynamics associated with different bifurcations of periodic and quasi-periodic regimes. As a result of these, two types of chaotic dynamics with one and two positive Lyapunov exponents are observed. We argue that the results obtained are robust as they can exist in a wide range of the system parameters

Added: Dec 2, 2019
Article
Bizyaev I. A., Borisov A. V., Mamaev I. S. Regular and Chaotic Dynamics. 2016. Vol. 21. No. 1. P. 136-146.

This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.

Added: Apr 5, 2017
Article
Bizyaev I. A., Borisov A. V., Kazakov A. Regular and Chaotic Dynamics. 2015. Vol. 20. No. 5. P. 605-626.

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with

regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.

Added: Oct 22, 2015
Article
Grines V., Malyshev D., Pochinka O. et al. Regular and Chaotic Dynamics. 2016. Vol. 21. No. 2. P. 189-203.

It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration algorithm. However, an efficient algorithm (i. e., polynomial in the number of vertices) has not yet been developed for it, and the problem has not been proved to be intractable (i. e., NP-complete). We give polynomial-time algorithms for recognition of the corresponding graphs for two gradient-like systems. Moreover, we present efficient algorithms for determining the orientability and the genus of the ambient surface. This result, in particular, sheds light on the classification of configurations that arise from simple, point-source potential-field models in efforts to determine the nature of the quiet-Sun magnetic field.

Added: Apr 5, 2016
Article
Bizyaev I. A., Borisov A. V., Killin A. A. et al. Regular and Chaotic Dynamics. 2016. Vol. 21. No. 6. P. 759-774.

This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincar´e map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.

Added: Apr 5, 2017
Article
Demina M.V., Kudryashov N. A. Regular and Chaotic Dynamics. 2016. Vol. 21. No. 3. P. 351-366.

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived.

Added: Oct 5, 2018
Article
Garashchuk I., Sinelshchikov D., Kudryashov N. A. Regular and Chaotic Dynamics. 2018. Vol. 23. P. 257-272.

Contrast agent microbubbles, which are encapsulated gas bubbles, are widely used to enhance ultrasound imaging. There are also several new promising applications of the contrast agents such as targeted drug delivery and noninvasive therapy. Here we study three models of the microbubble dynamics: a nonencapsulated bubble oscillating close to an elastic wall, a simple coated bubble and a coated bubble near an elastic wall.We demonstrate that complex dynamics can occur in these models. We are particularly interested in the multistability phenomenon of bubble dynamics. We show that coexisting attractors appear in all of these models, but for higher acoustic pressures for the models of an encapsulated bubble.We demonstrate how several tools can be used to localize the coexisting attractors. We provide some considerations why the multistability can be undesirable for applications.

Added: Dec 16, 2019
Article
Pochinka O., Levchenko Y., Grines V. et al. Regular and Chaotic Dynamics. 2014. Vol. 19. No. 4. P. 506-512.

We prove that each structurally stable diffeomorphism $f$ on a closed 3-manifold $M^3$ with two-dimensional surface nonwandering set is topologically conjugated to some model dynamically coherent diffeomorphism.

Added: Sep 11, 2014
Article
Burov A. A., Shalimova E. Regular and Chaotic Dynamics. 2015. Vol. 20. No. 3. P. 225-233.

The problem of motion of a heavy particle on a sphere uniformly rotating about a fixed axis is considered in the case of dry friction. It is assumed that the angle of inclination of the rotation axis is constant. The existence of equilibria in an absolute coordinate system and their linear stability are discussed. The equilibria in a relative coordinate system rotating with the sphere are also studied. These equilibria are generally nonisolated. The dependence of the equilibrium sets of this kind on the system parameters is also considered. © 2015, Pleiades Publishing, Ltd.

Added: Sep 3, 2015
Article
Grines V., Gurevich E., Pochinka O. Regular and Chaotic Dynamics. 2017. Vol. 22. No. 2. P. 122-135.

Separators are fundamental plasma physics objects that play an important role in many astrophysical phenomena. Looking for separators and their number is one of the first steps in studying the topology of the magnetic field in the solar corona. In the language of dynamical systems, separators are noncompact heteroclinic curves. In this paper we give an exact lower estimation of the number of noncompact heteroclinic curves for a 3-diffeomorphism with the so-called “surface dynamics”. Also, we prove that ambient manifolds for such diffeomorphisms are mapping tori. 

Added: May 11, 2017
Article
Kruglov V., Malyshev D., Pochinka O. et al. Regular and Chaotic Dynamics. 2020. Vol. 25. No. 6. P. 716-728.

In this paper, we study gradient-like flows without heteroclinic intersections on n-sphere up to topological conjugacy. We prove that such a flow is completely defined by a bi-colour tree corresponding to a skeleton formed by co-dimension one separatrices. Moreover, we show that such a tree is a complete invariant for these flows with respect to the topological equivalence also. The obtained result means that for considered flows with the same (up to a change of coordinates) partitions on trajectories, the partitions for elements, composing isotopies connecting time-one shifts of these flows with the identity map, also coincide. This phenomena strongly contrasts with the situation for flows with periodic orbits and connections, where one class of the equivalence contains continuum classes of the conjugacy. Additionally, we realize every connected bi-colour tree by a gradient-like flow without heteroclinic intersections on n-sphere. Besides, we present a linear-time algorithm on the number of vertices for distinguishing these trees.

Added: Nov 15, 2020
Article
Kazakov A., Борисов А. В., Пивоварова Е. Н. Regular and Chaotic Dynamics. 2016. Vol. 21. No. 7-8. P. 885-901.
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of period-doubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.
Added: Jan 30, 2017
Article
Гонченко А. С., Гонченко С. В., Kazakov A. Regular and Chaotic Dynamics. 2013. Vol. 18. No. 5. P. 521-538.

We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.

Added: Mar 29, 2015
Article
Kazakov A., Korotkov A., Osipov G. V. Regular and Chaotic Dynamics. 2015. Vol. 20. No. 6. P. 701-715.

In this article a new model of motif (small ensemble) of neuron-like elements is proposed. It is built with the use of generalized Lotka-Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of the brain. In this paper it is shown that there are two modes depending on the type of coupling between the elements: the mode with a stable heteroclinic cycle and the mode with a stable limit cycle. Our second goal is to examine the chaotic dynamics of generalized three-dimensional Lotka-Volterra model.

Added: Oct 22, 2015
Article
Борисов А. В., Kazakov A., Сатаев И. Р. Regular and Chaotic Dynamics. 2016. Vol. 21. No. 7-8. P. 939-954.

This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a three-dimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn about the influence of “strangeness” of the attractor on the motion pattern of the top.

Added: Nov 21, 2016
Article
Safonova D. V., Demina M.V., Kudryashov N. A. Regular and Chaotic Dynamics. 2018. Vol. 23. No. 5. P. 569-579.

In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation $\Gamma_1$ and $\Gamma_2$ are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained.

Added: Oct 26, 2018
Article
Kazakov A. Regular and Chaotic Dynamics. 2013. Vol. 18. No. 5. P. 508-520.

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The termrubbermeans that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincar´emapis enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.

Added: Mar 29, 2015
Article
Kazakov A., Borisov A. V., Sataev I. R. Regular and Chaotic Dynamics. 2014. Vol. 19. No. 6. P. 718-733.
In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.
Added: Mar 29, 2015
Article
Borisov A. V., Mamaev I. S., Bizyaev I. A. Regular and Chaotic Dynamics. 2016. Vol. 21. No. 5. P. 556-580.

In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincar´e section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems.

Added: Apr 5, 2017
Article
Ilyashenko Y. Regular and Chaotic Dynamics. 2010. No. 15(2-3). P. 328-334.
Added: Feb 15, 2012