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2013
Impact Factor

    Alexey Kazakov

    Alexey Kazakov
    Universitetskaya 1, Izhevsk, 426034 Russia
    Institute of Computer Science, Udmurt State University

    Publications:

    Borisov A. V., Kazakov A. O., Pivovarova E. N.
    Abstract
    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 perioddoubling 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.
    Keywords: Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact
    Citation: Borisov A. V., Kazakov A. O., Pivovarova E. N.,  Regular and chaotic dynamics in the rubber model of a Chaplygin top, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 2, pp.  277-297
    DOI:10.20537/nd1702009
    Bizyaev I. A., Borisov A. V., Kazakov A. O.
    Abstract
    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.
    Keywords: Suslov problem, nonholonomic constraint, reversal, strange attractor
    Citation: Bizyaev I. A., Borisov A. V., Kazakov A. O.,  Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 2, pp.  263-287
    DOI:10.20537/nd1602008
    Sataev I. R., Kazakov A. O.
    Abstract
    We study the dynamics in the Suslov problem which describes 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) motions and, using a new method for constructing charts of Lyapunov exponents, detect different types of chaotic behavior such as conservative chaos, strange attractors and mixed dynamics, which are typical of reversible systems. In the paper we also examine the phenomenon of reversal, which was observed previously in the motion of Celtic stones.
    Keywords: nonholonomic model, Chaplygin top, Afraimovich – Shilnikov torus-breakdown, cascade of period-doubling bifurcations, scenario of period doublings of tori, figure-eight attractor
    Citation: Sataev I. R., Kazakov A. O.,  Scenarios of transition to chaos in the nonholonomic model of a Chaplygin top, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 2, pp.  235-250
    DOI:10.20537/nd1602006
    Vetchanin  E. V., Kazakov A. O.
    Abstract
    This paper is concerned with the dynamics of two point vortices of the same intensity which are affected by an acoustic wave. Typical bifurcations of fixed points have been identified by constructing charts of dynamical regimes, and bifurcation diagrams have been plotted.
    Keywords: point vortices, nonintegrability, bifurcations, chart of dynamical regimes
    Citation: Vetchanin  E. V., Kazakov A. O.,  Bifurcations and chaos in the problem of the motion of two point vortices in an acoustic wave, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 3, pp.  329-343
    DOI:10.20537/nd1403007
    Borisov A. V., Kazakov A. O., Sataev I. R.
    Abstract
    We study both analytically and numerically the dynamics of an inhomogeneous ball on a rough horizontal plane under the infuence of gravity. A nonholonomic constraint of zero velocity at the point of contact of the ball with the plane is imposed. In the case of an arbitrary displacement of the center of mass of the ball, the system is nonintegrable without the property of phase volume conservation. We show that at certain parameter values the unbalanced ball exhibits the effect of reversal (the direction of the ball rotation reverses). Charts of dynamical regimes on the parameter plane are presented. The system under consideration exhibits diverse chaotic dynamics, in particular, the figure-eight chaotic attractor, which is a special type of pseudohyperbolic chaos.
    Keywords: Chaplygin’s top, rolling without slipping, reversibility, involution, integrability, reverse, chart of dynamical regimes, strange attractor
    Citation: Borisov A. V., Kazakov A. O., Sataev I. R.,  Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 3, pp.  361-380
    DOI:10.20537/nd1403010
    Bolsinov A. V., Kilin A. A., Kazakov A. O.
    Topological monodromy in nonholonomic systems
    2013, Vol. 9, No. 2, pp.  203-227
    Abstract
    The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.
    Keywords: topological monodromy, integrable systems, nonholonomic systems, Poincaré map, bifurcation analysis, focus-focus singularities
    Citation: Bolsinov A. V., Kilin A. A., Kazakov A. O.,  Topological monodromy in nonholonomic systems, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 2, pp.  203-227
    DOI:10.20537/nd1302002
    Bizyaev I. A., Kazakov A. O.
    Abstract
    In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of an ellipsoid on a plane and a sphere. We research these problems using Poincare maps, which investigation helps to discover a new integrable case.
    Keywords: nonholonomic constraint, invariant measure, first integral, Poincare map, integrability and chaos
    Citation: Bizyaev I. A., Kazakov A. O.,  Integrability and stochastic behavior in some nonholonomic dynamics problems, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 2, pp.  257-265
    DOI:10.20537/nd1302005
    Kazakov A. O.
    Abstract
    In this paper we study a problem of rolling of the dynamically asymmetric ball with displacement center of gravity on a plane without slipping and vertical rotating. It is shown that the dynamics of the ball is significantly affected by the type of reversibility. Depending on the type of the reversibility we found two different types of dynamical chaos: strange attractors and mixed chaotic dynamics. In this paper we describe a strange attractor development, and then its basic properties. A set of criteria by which in numerical experiments mixed dynamics may be distinguished from other types of dynamical chaos are given.
    Keywords: rock-n-roller, rubber rolling, reversibility, bifurcation, focus, saddle, separatrix, homoclinic tangency, Lyapunov’s exponents, mixed dynamics, strange attractor
    Citation: Kazakov A. O.,  Chaotic dynamics phenomena in the rubber rock-n-roller on a plane problem, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 2, pp.  309-325
    DOI:10.20537/nd1302008
    Gonchenko A. S., Gonchenko S. V., Kazakov A. O.
    On some new aspects of Celtic stone chaotic dynamics
    2012, Vol. 8, No. 3, pp.  507-518
    Abstract
    We study chaotic dynamics of a nonholonomic model of celtic stone movement on the plane. Scenarious of appearance and development of chaos are investigated.
    Keywords: nonholonomic model, strange attractor, symmetry, bifurcation, mixed dynamics
    Citation: Gonchenko A. S., Gonchenko S. V., Kazakov A. O.,  On some new aspects of Celtic stone chaotic dynamics, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp.  507-518
    DOI:10.20537/nd1203006

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