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2013
Impact Factor

# Elena Pivovarova

Universitetskaya, 1, Izhevsk, 426034, Russia
Udmurt State University

## Publications:

 Kilin A. A., Pivovarova E. N. A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane 2021, vol. 26, no. 6, pp.  775-786 Abstract In this paper we investigate the motion of a Chaplygin sphere rolling without slipping on a plane performing horizontal periodic oscillations. We show that in the system under consideration the projections of the angular momentum onto the axes of the fixed coordinate system remain unchanged. The investigation of the reduced system on a fixed level set of first integrals reduces to analyzing a three-dimensional period advance map on $SO(3)$. The analysis of this map suggests that in the general case the problem considered is nonintegrable. We find partial solutions to the system which are a generalization of permanent rotations and correspond to nonuniform rotations about a body- and space-fixed axis. We also find a particular integrable case which, after time is rescaled, reduces to the classical Chaplygin sphere rolling problem on the zero level set of the area integral. Keywords: Chaplygin sphere, rolling motion, nonholonomic constraint, nonautonomous dynamical system, periodic oscillations, permanent rotations, integrable case, period advance map Citation: Kilin A. A., Pivovarova E. N.,  A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 775-786 DOI:10.1134/S1560354721060149
 Kilin A. A., Pivovarova E. N. Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges 2019, vol. 24, no. 2, pp.  212-233 Abstract This paper presents a qualitative analysis of the dynamics in a fixed reference frame of a wheel with sharp edges that rolls on a horizontal plane without slipping at the point of contact and without spinning relative to the vertical. The wheel is a ball that is symmetrically truncated on both sides and has a displaced center of mass. The dynamics of such a system is described by the model of the ball’s motion where the wheel rolls with its spherical part in contact with the supporting plane and the model of the disk’s motion where the contact point lies on the sharp edge of the wheel. A classification is given of possible motions of the wheel depending on whether there are transitions from its spherical part to sharp edges. An analysis is made of the behavior of the point of contact of the wheel with the plane for different values of the system parameters, first integrals and initial conditions. Conditions for boundedness and unboundedness of the wheel’s motion are obtained. Conditions for the fall of the wheel on the plane of sections are presented. Keywords: integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber body model, permanent rotations, dynamics in a fixed reference frame, resonance, quadrature, unbounded motion Citation: Kilin A. A., Pivovarova E. N.,  Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 212-233 DOI:10.1134/S1560354719020072
 Kilin A. A., Pivovarova E. N. Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges 2018, vol. 23, no. 7-8, pp.  887-907 Abstract This paper is concerned with the dynamics of a wheel with sharp edges moving on a horizontal plane without slipping and rotation about the vertical (nonholonomic rubber model). The wheel is a body of revolution and has the form of a ball symmetrically truncated on both sides. This problem is described by a system of differential equations with a discontinuous right-hand side. It is shown that this system is integrable and reduces to quadratures. Partial solutions are found which correspond to fixed points of the reduced system. A bifurcation analysis and a classification of possible types of the wheel’s motion depending on the system parameters are presented. Keywords: integrable system, system with a discontinuous right-hand side, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber model Citation: Kilin A. A., Pivovarova E. N.,  Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 887-907 DOI:10.1134/S1560354718070067
 Kilin A. A., Pivovarova E. N. The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane 2017, vol. 22, no. 3, pp.  298-317 Abstract This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded. Keywords: integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, absolute dynamics Citation: Kilin A. A., Pivovarova E. N.,  The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 298-317 DOI:10.1134/S156035471703008X
 Borisov A. V., Kazakov A. O., Pivovarova E. N. Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top 2016, vol. 21, no. 7-8, pp.  885-901 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 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. 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, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901 DOI:10.1134/S156035471607011X
 Kilin A. A., Pivovarova E. N., Ivanova T. B. Spherical Robot of Combined Type: Dynamics and Control 2015, vol. 20, no. 6, pp.  716-728 Abstract This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic model are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary trajectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point. Keywords: spherical robot, control, nonholonomic constraint, combined mechanism Citation: Kilin A. A., Pivovarova E. N., Ivanova T. B.,  Spherical Robot of Combined Type: Dynamics and Control, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 716-728 DOI:10.1134/S1560354715060076
 Ivanova T. B., Pivovarova E. N. Comments on the Paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev "How to Control the Chaplygin Ball Using Rotors. II" 2014, vol. 19, no. 1, pp.  140-143 Abstract In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed. Keywords: control, dry friction, Chaplygin’s ball, spherical robot Citation: Ivanova T. B., Pivovarova E. N.,  Comments on the Paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev "How to Control the Chaplygin Ball Using Rotors. II", Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 140-143 DOI:10.1134/S1560354714010092