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2013
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# Alexander Kilin

1, Universitetskaya str., Izhevsk 426034, Russia
aka@rcd.ru
Institute of Computer Science

Dean of the Faculty of Physics and Energetics,Udmurt State University

Senior Scientist

Institute of Computer Science and  Department of Mathematical Methods in Nonlinear Dynamics, IMM UB RAS

Born: May 31, 1976
In 1997 graduated from Udmurt State University (UdSU).
1997-2001: research assistant in Laboratory of dynamical Chaos and Nonlinearity, UdSU.
2001: Thesis of Ph.D. (candidate of science). Thesis title: "Computer-aided methods in study of nonlinear dynamical systems", UdSU
2002: senior scientist of Laboratory of Dynamical Chaos an Nonlinearity, UdSU.
2004-present:- Senior scientist of Department of Mathematical Methods in Nonlinear Dynamics, IMM UB RAS;
-Scientific secretary of Institute of Computer Science
2009: Doctor in physics and mathematics. Thesis title: "Development of the software package for computer studies of dynamical systems", Moscow Engineering Physics Institute.
since 2010: Head of Laboratory of Dynamical Chaos and Nonlinearity at UdSU
since 2011: Dean of the Faculty of Physics and Energetics at UdSU

Jair Koiller (Getulio Vargas Foundation Graduate School of Economics, Brazil)

## Publications:

 Borisov A. V., Kilin A. A., Mamaev I. S. A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness 2019, vol. 24, no. 3, pp.  329-352 Abstract This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both well-known results and a number of new results. We consider the two most widespread friction (resistance) models: dry (Coulomb) friction and viscous friction. It is shown that the addition of external damping (air drag) can lead to stability of equilibrium at the saddle point and hence to preservation of the region of bounded motion in a neighborhood of the saddle point. Analysis of three-dimensional Poincaré sections shows that limit cycles can arise in this case in the neighborhood of the saddle point. Keywords: parabolic pendulum, Paul trap, rotating paraboloid, internal damping, external damping, friction, resistance, linear stability, Hill’s region, bifurcational diagram, Poincaré section, bounded trajectory, chaos, integrability, nonintegrability, sepa Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 329-352 DOI:10.1134/S1560354719030067
 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., Artemova E. M. Integrability and Chaos in Vortex Lattice Dynamics 2019, vol. 24, no. 1, pp.  101-113 Abstract This paper is concerned with the problem of the interaction of vortex lattices, which is equivalent to the problem of the motion of point vortices on a torus. It is shown that the dynamics of a system of two vortices does not depend qualitatively on their strengths. Steadystate configurations are found and their stability is investigated. For two vortex lattices it is also shown that, in absolute space, vortices move along closed trajectories except for the case of a vortex pair. The problems of the motion of three and four vortex lattices with nonzero total strength are considered. For three vortices, a reduction to the level set of first integrals is performed. The nonintegrability of this problem is numerically shown. It is demonstrated that the equations of motion of four vortices on a torus admit an invariant manifold which corresponds to centrally symmetric vortex configurations. Equations of motion of four vortices on this invariant manifold and on a fixed level set of first integrals are obtained and their nonintegrability is numerically proved. Keywords: vortices on a torus, vortex lattices, point vortices, nonintegrability, chaos, invariant manifold, Poincarґe map, topological analysis, numerical analysis, accuracy of calculations, reduction, reduced system Citation: Kilin A. A., Artemova E. M.,  Integrability and Chaos in Vortex Lattice Dynamics, Regular and Chaotic Dynamics, 2019, vol. 24, no. 1, pp. 101-113 DOI:10.1134/S1560354719010064
 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
 Borisov A. V., Kilin A. A., Mamaev I. S. A Nonholonomic Model of the Paul Trap 2018, vol. 23, no. 3, pp.  339-354 Abstract In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the hyperbolic paraboloid is made. A three-dimensional Poincar´e map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case. Keywords: Paul trap, stability, nonholonomic system, three-dimensional map, gyroscopic stabilization, noninertial coordinate system, Poincaré map, nonholonomic constraint, rolling without slipping, region of linear stability Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  A Nonholonomic Model of the Paul Trap, Regular and Chaotic Dynamics, 2018, vol. 23, no. 3, pp. 339-354 DOI:10.1134/S1560354718030085
 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
 Vetchanin  E. V., Kilin A. A., Mamaev I. S. Control of the Motion of a Helical Body in a Fluid Using Rotors 2016, vol. 21, no. 7-8, pp.  874-884 Abstract This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of sub-Riemannian geodesics on $SE(3)$ are obtained. Keywords: ideal fluid, motion of a helical body, Kirchhoff equations, control of rotors, gaits, optimal control Citation: Vetchanin  E. V., Kilin A. A., Mamaev I. S.,  Control of the Motion of a Helical Body in a Fluid Using Rotors, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 874-884 DOI:10.1134/S1560354716070108
 Karavaev Y. L., Kilin A. A., Klekovkin A. V. Experimental Investigations of the Controlled Motion of a Screwless Underwater Robot 2016, vol. 21, no. 7-8, pp.  918-926 Abstract In this paper we describe the results of experimental investigations of the motion of a screwless underwater robot controlled by rotating internal rotors. We present the results of comparison of the trajectories obtained with the results of numerical simulation using the model of an ideal fluid. Keywords: screwless underwater robot, experimental investigations, helical body Citation: Karavaev Y. L., Kilin A. A., Klekovkin A. V.,  Experimental Investigations of the Controlled Motion of a Screwless Underwater Robot, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 918-926 DOI:10.1134/S1560354716070133
 Klenov A. I., Kilin A. A. Influence of Vortex Structures on the Controlled Motion of an Above-water Screwless Robot 2016, vol. 21, no. 7-8, pp.  927-938 Abstract This paper is devoted to an experimental investigation of the motion of a rigid body set in motion by rotating two unbalanced internal masses. The results of experiments confirming the possibility of motion by this method are presented. The dependence of the parameters of motion on the rotational velocity of internal masses is analyzed. The velocity field of the fluid around the moving body is examined. Keywords: self-propulsion, PIV, vortex formation, above-water screwless robot Citation: Klenov A. I., Kilin A. A.,  Influence of Vortex Structures on the Controlled Motion of an Above-water Screwless Robot, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 927-938 DOI:10.1134/S1560354716070145
 Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S. Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups 2016, vol. 21, no. 6, pp.  759-774 Abstract 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 Poincaré 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. Keywords: sub-Riemannian geometry, Carnot group, Poincaré map, first integrals Citation: Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.,  Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 759-774 DOI:10.1134/S1560354716060125
 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
 Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A. Qualitative Analysis of the Dynamics of a Wheeled Vehicle 2015, vol. 20, no. 6, pp.  739-751 Abstract This paper is concerned with the problem of the motion of a wheeled vehicle on a plane in the case where one of the wheel pairs is fixed. In addition, the motion of a wheeled vehicle on a plane in the case of two free wheel pairs is considered. A method for obtaining equations of motion for the vehicle with an arbitrary geometry is presented. Possible kinds of motion of the vehicle with a fixed wheel pair are determined. Keywords: nonholonomic constraint, system dynamics, wheeled vehicle, Chaplygin system Citation: Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A.,  Qualitative Analysis of the Dynamics of a Wheeled Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 739-751 DOI:10.1134/S156035471506009X
 Borisov A. V., Kilin A. A., Mamaev I. S. On the Hadamard – Hamel Problem and the Dynamics of Wheeled Vehicles 2015, vol. 20, no. 6, pp.  752-766 Abstract In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs. Keywords: nonholonomic constraint, wheeled vehicle, reduction, equations of motion Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  On the Hadamard – Hamel Problem and the Dynamics of Wheeled Vehicles, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 752-766 DOI:10.1134/S1560354715060106
 Karavaev Y. L., Kilin A. A. The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform 2015, vol. 20, no. 2, pp.  134-152 Abstract This paper deals with the problem of a spherical robot propelled by an internal omniwheel platform and rolling without slipping on a plane. The problem of control of spherical robot motion along an arbitrary trajectory is solved within the framework of a kinematic model and a dynamic model. A number of particular cases of motion are identified, and their stability is investigated. An algorithm for constructing elementary maneuvers (gaits) providing the transition from one steady-state motion to another is presented for the dynamic model. A number of experiments have been carried out confirming the adequacy of the proposed kinematic model. Keywords: spherical robot, kinematic model, dynamic model, nonholonomic constraint, omniwheel Citation: Karavaev Y. L., Kilin A. A.,  The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 134-152 DOI:10.1134/S1560354715020033
 Borisov A. V., Kilin A. A., Mamaev I. S. Dynamics and Control of an Omniwheel Vehicle 2015, vol. 20, no. 2, pp.  153-172 Abstract A nonholonomic model of the dynamics of an omniwheel vehicle on a plane and a sphere is considered. A derivation of equations is presented and the dynamics of a free system are investigated. An explicit motion control algorithm for the omniwheel vehicle moving along an arbitrary trajectory is obtained. Keywords: omniwheel, roller-bearing wheel, nonholonomic constraint, dynamical system, invariant measure, integrability, controllability Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Dynamics and Control of an Omniwheel Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 153-172 DOI:10.1134/S1560354715020045
 Borisov A. V., Kilin A. A., Mamaev I. S. The Problem of Drift and Recurrence for the Rolling Chaplygin Ball 2013, vol. 18, no. 6, pp.  832-859 Abstract We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point. Keywords: nonholonomic constraint, absolute dynamics, bifurcation diagram, bifurcation complex, drift, resonance, invariant torus Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 832-859 DOI:10.1134/S1560354713060166
 Borisov A. V., Kilin A. A., Mamaev I. S. The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem 2013, vol. 18, no. 1-2, pp.  33-62 Abstract We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings. Keywords: ideal fluid, vortex ring, leapfrogging motion of vortex rings, bifurcation complex, periodic solution, integrability, chaotic dynamics Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 33-62 DOI:10.1134/S1560354713010036
 Borisov A. V., Kilin A. A., Mamaev I. S. How to Control the Chaplygin Ball Using Rotors. II 2013, vol. 18, no. 1-2, pp.  144-158 Abstract In our earlier paper [3] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed. Keywords: non-holonomic constraint, control, dry friction, viscous friction, stability, periodic solutions Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  How to Control the Chaplygin Ball Using Rotors. II, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 144-158 DOI:10.1134/S1560354713010103
 Borisov A. V., Kilin A. A., Mamaev I. S. How to Control Chaplygin’s Sphere Using Rotors 2012, vol. 17, no. 3-4, pp.  258-272 Abstract In the paper we study the control of a balanced dynamically non-symmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic controllability is shown and the control inputs that steer the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed. Keywords: non-holonomic constraint, non-holonomic distribution, control, Chow–Rashevsky theorem, drift Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  How to Control Chaplygin’s Sphere Using Rotors, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 258-272 DOI:10.1134/S1560354712030045
 Borisov A. V., Kilin A. A., Mamaev I. S. Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support 2012, vol. 17, no. 2, pp.  170-190 Abstract We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu.N.Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support – the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability. Keywords: nonholonomic mechanics, spherical support, Chaplygin ball, explicit integration, isomorphism, bifurcation analysis Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 170-190 DOI:10.1134/S1560354712020062
 Borisov A. V., Kilin A. A., Mamaev I. S. On the Model of Non-holonomic Billiard 2011, vol. 16, no. 6, pp.  653-662 Abstract In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown. Keywords: billiard, impact, point map, nonintegrability, periodic solution, nonholonomic constraint, integral of motion Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  On the Model of Non-holonomic Billiard, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 653-662 DOI:10.1134/S1560354711060062
 Borisov A. V., Kilin A. A., Mamaev I. S. Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere 2011, vol. 16, no. 5, pp.  465-483 Abstract We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of "clandestine" linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability. Keywords: nonholonomic constraint, rolling motion, Chaplygin ball, integral, invariant measure Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 465-483 DOI:10.1134/S1560354711050042
 Borisov A. V., Kilin A. A., Mamaev I. S. Hamiltonicity and integrability of the Suslov problem 2011, vol. 16, no. 1-2, pp.  104-116 Abstract The Hamiltonian representation and integrability of the nonholonomic Suslov problem and its generalization suggested by S. A. Chaplygin are considered. This subject is important for understanding the qualitative features of the dynamics of this system, being in particular related to a nontrivial asymptotic behavior (i. e., to a certain scattering problem). A general approach based on studying a hierarchy in the dynamical behavior of nonholonomic systems is developed. Keywords: Hamiltonian system, Poisson bracket, nonholonomic constraint, invariant measure, integrability Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Hamiltonicity and integrability of the Suslov problem, Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 104-116 DOI:10.1134/S1560354711010035
 Borisov A. V., Kilin A. A., Mamaev I. S. Superintegrable system on a sphere with the integral of higher degree 2009, vol. 14, no. 6, pp.  615-620 Abstract We consider the motion of a material point on the surface of a sphere in the field of $2n + 1$ identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [1], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional $N$-particle system discussed in the recent paper [2] and show that for the latter system an analogous superintegral can be constructed. Keywords: superintegrable systems, systems with a potential, Hooke center Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Superintegrable system on a sphere with the integral of higher degree, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 615-620 DOI:10.1134/S156035470906001X
 Borisov A. V., Mamaev I. S., Kilin A. A. The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids 2009, vol. 14, no. 2, pp.  179-217 Abstract The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors’ original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices). Keywords: liquid and gas self-gravitating ellipsoids, integrability, chaotic behavior Citation: Borisov A. V., Mamaev I. S., Kilin A. A.,  The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids, Regular and Chaotic Dynamics, 2009, vol. 14, no. 2, pp. 179-217 DOI:10.1134/S1560354709020014
 Borisov A. V., Kilin A. A., Mamaev I. S. Multiparticle Systems. The Algebra of Integrals and Integrable Cases 2009, vol. 14, no. 1, pp.  18-41 Abstract Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector. A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a particle­interaction potential homogeneous of degree $\alpha = –2$ are considered. The most general form of the additional integral of motion, which we term the Jacobi integral, is presented for such systems. A new nonlinear algebra of integrals including the Jacobi integral is found. A systematic description is given to a new reduction procedure and possibilities of applying it to dynamics with the aim of lowering the order of Hamiltonian systems. Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations of the Lagrangian identity for systems with a particle­ interaction potential homogeneous of degree $\alpha = –2$ are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane. Keywords: multiparticle systems, Jacobi integral Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Multiparticle Systems. The Algebra of Integrals and Integrable Cases, Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 18-41 DOI:10.1134/S1560354709010043
 Borisov A. V., Kilin A. A., Mamaev I. S. Stability of Steady Rotations in the Nonholonomic Routh Problem 2008, vol. 13, no. 4, pp.  239-249 Abstract We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point. Keywords: nonholonomic constraint, stationary rotations, stability Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Stability of Steady Rotations in the Nonholonomic Routh Problem, Regular and Chaotic Dynamics, 2008, vol. 13, no. 4, pp. 239-249 DOI:10.1134/S1560354708040011
 Borisov A. V., Kilin A. A., Mamaev I. S. Absolute and Relative Choreographies in Rigid Body Dynamics 2008, vol. 13, no. 3, pp.  204-220 Abstract For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev–Chaplygin case, and the Steklov solution. The "genealogy" of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies). Keywords: rigid-body dynamics, periodic solutions, continuation by a parameter, bifurcation Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Absolute and Relative Choreographies in Rigid Body Dynamics, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 204-220 DOI:10.1134/S1560354708030064
 Borisov A. V., Kilin A. A., Mamaev I. S. Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point 2008, vol. 13, no. 3, pp.  221-233 Abstract In this paper, we consider the transition to chaos in the phase portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotization are indicated: (1) the growth of the homoclinic structure and (2) the development of cascades of period doubling bifurcations. On the zero level of the area integral, an adiabatic behavior of the system (as the energy tends to zero) is noted. Meander tori induced by the break of the torsion property of the mapping are found. Keywords: motion of a rigid body, phase portrait, mechanism of chaotization, bifurcations Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 221-233 DOI:10.1134/S1560354708030076
 Borisov A. V., Mamaev I. S., Kilin A. A. Two-body problem on a sphere. Reduction, stochasticity, periodic orbits 2004, vol. 9, no. 3, pp.  265-279 Abstract We consider the problem of two interacting particles on a sphere. The potential of the interaction depends on the distance between the particles. The case of Newtonian-type potentials is studied in most detail. We reduce this system to a system with two degrees of freedom and give a number of remarkable periodic orbits. We also discuss integrability and stochastization of the motion. Citation: Borisov A. V., Mamaev I. S., Kilin A. A.,  Two-body problem on a sphere. Reduction, stochasticity, periodic orbits, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 265-279 DOI:10.1070/RD2004v009n03ABEH000280
 Borisov A. V., Mamaev I. S., Kilin A. A. Absolute and relative choreographies in the problem of point vortices moving on a plane 2004, vol. 9, no. 2, pp.  101-111 Abstract We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the $n$-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features. Citation: Borisov A. V., Mamaev I. S., Kilin A. A.,  Absolute and relative choreographies in the problem of point vortices moving on a plane, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 101-111 DOI:10.1070/RD2004v009n02ABEH000269
 Borisov A. V., Mamaev I. S., Kilin A. A. Dynamics of rolling disk 2003, vol. 8, no. 2, pp.  201-212 Abstract In the paper we present the qualitative analysis of rolling motion without slipping of a homogeneous round disk on a horisontal plane. The problem was studied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its integrability. The behavior of the point of contact on a plane is investigated and conditions under which its trajectory is finit are obtained. The bifurcation diagrams are constructed. Citation: Borisov A. V., Mamaev I. S., Kilin A. A.,  Dynamics of rolling disk, Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 201-212 DOI:10.1070/RD2003v008n02ABEH000237
 Borisov A. V., Mamaev I. S., Kilin A. A. The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics 2002, vol. 7, no. 2, pp.  201-219 Abstract The paper is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The paper also indicates a special case when an additional integral and invariant measure exist. Using this case, we obtain a nonholonomic generalization of the Jacobi problem for the inertial motion of a point on an ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder in the gravity field the ball's motion is bounded and, on the average, it does not move downwards. All the results of the paper considerably expand the results obtained by E. Routh in XIX century. Citation: Borisov A. V., Mamaev I. S., Kilin A. A.,  The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 201-219 DOI:10.1070/RD2002v007n02ABEH000205
 Kilin A. A. The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis 2001, vol. 6, no. 3, pp.  291-306 Abstract The motion of Chaplygin ball with and without gyroscope in the absolute space is analyzed. In particular, the trajectories of the point of contact are studied in detail. We discuss the motions in the absolute space, that correspond to the different types of motion in the moving frame of reference related to the body. The existence of the bounded trajectories of the ball's motion is shown by means of numerical methods in the case when the problem is reduced to a certain Hamiltonian system. Citation: Kilin A. A.,  The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 291-306 DOI:10.1070/RD2001v006n03ABEH000178
 Kilin A. A. First Integral in the Problem of Motion of a Circular Cylinder and a Point Vortex in Unbounded Ideal Fluid 2001, vol. 6, no. 2, pp.  233-234 Abstract In the paper Motion of a circular cylinder and a vortex in an ideal fluid (Reg. & Chaot. Dyn. V. 6. 2001. No 1. P. 33-38) Ramodanov S.M. showed the integrability of the problem of motion of a circular cylinder and a point vortex in unbounded ideal fluid. In the present paper we find additional first integral and invariant measure of motion equations. Citation: Kilin A. A.,  First Integral in the Problem of Motion of a Circular Cylinder and a Point Vortex in Unbounded Ideal Fluid, Regular and Chaotic Dynamics, 2001, vol. 6, no. 2, pp. 233-234 DOI:10.1070/RD2001v006n02ABEH000174
 Borisov A. V., Kilin A. A. Stability of Thomson's Configurations of Vortices on a Sphere 2000, vol. 5, no. 2, pp.  189-200 Abstract In this work stability of polygonal configurations on a plane and sphere is investigated. The conditions of linear stability are obtained. A nonlinear analysis of the problem is made with the help of Birkhoff normalization. Some problems are also formulated. Citation: Borisov A. V., Kilin A. A.,  Stability of Thomson's Configurations of Vortices on a Sphere, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 189-200 DOI:10.1070/RD2000v005n02ABEH000141
 Kilin A. A. Libration points in spaces $S^2$ and $L^2$ 1999, vol. 4, no. 1, pp.  91-103 Abstract We consider two-body problem and restricted three-body problem in spaces $S^2$ and $L^2$. For two-body problem we have showed the absence of exponential instability of partiбular solutions relevant to roundabout motion on the plane. New libration points are found, and the dependence of their positions on parameters of a system is explored. The regions of existence of libration points in space of parameters were constructed. Basing on a examination of the Hill's regions we found the qualitative estimation of stability of libration points was produced. Citation: Kilin A. A.,  Libration points in spaces $S^2$ and $L^2$, Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 91-103 DOI:10.1070/RD1999v004n01ABEH000101