Alexander Kilin
Doctor of Physics and Mathematics, Professor
Professor of Department of Theoretical Physics at UdSU
Head of Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles at UdSU
Leading Researcher of the Department of Mathematical Methods of Nonlinear Dynamics at Institute of Mathematics and Mechanics UB RAS
Born: May 31, 1976
In 1997 graduated from Udmurt State University (UdSU).
19972001: research assistant in Laboratory of dynamical Chaos and Nonlinearity, UdSU.
2001: Thesis of Ph.D. (candidate of science). Thesis title: "Computeraided methods in study of nonlinear dynamical systems", UdSU
2002: senior scientist of Laboratory of Dynamical Chaos an Nonlinearity, UdSU.
2004present: 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
Publications:
Kilin A. A., Pivovarova E. N.
Dynamics of an Unbalanced Disk with a Single Nonholonomic Constraint
2023, vol. 28, no. 1, pp. 78106
Abstract
The problem of the rolling of a disk on a plane is considered under the assumption
that there is no slipping in the direction parallel to the horizontal diameter of the disk and
that the center of mass does not move in the horizontal direction. This problem is reduced to
investigating a system of three firstorder differential equations. It is shown that the reduced
system is reversible relative to involution of codimension one and admits a twoparameter family
of fixed points. The linear stability of these fixed points is analyzed. Using numerical simulation,
the nonintegrability of the problem is shown. It is proved that the reduced system admits, even
in the nonintegrable case, a twoparameter family of periodic solutions. A number of dynamical
effects due to the existence of involution of codimension one and to the degeneracy of the fixed
points of the reduced system are found.

Artemova E. M., Kilin A. A.
Dynamics of Two Vortex Rings in a Bose – Einstein Condensate
2022, vol. 27, no. 6, pp. 713732
Abstract
In this paper, we consider the dynamics of two interacting point vortex rings in
a Bose – Einstein condensate. The existence of an invariant manifold corresponding to vortex
rings is proved. Equations of motion on this invariant manifold are obtained for an arbitrary
number of rings from an arbitrary number of vortices. A detailed analysis is made of the case
of two vortex rings each of which consists of two point vortices where all vortices have same
topological charge. For this case, partial solutions are found and a complete bifurcation analysis
is carried out. It is shown that, depending on the parameters of the Bose – Einstein condensate,
there are three different types of bifurcation diagrams. For each type, typical phase portraits
are presented.

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. 775786
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 threedimensional 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 spacefixed 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.

Kilin A. A., Pivovarova E. N.
Stability and Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base
2020, vol. 25, no. 6, pp. 729752
Abstract
This paper addresses the problem of a spherical robot having an axisymmetric pendulum drive and rolling without slipping on a vibrating plane. It is shown that this system admits partial solutions (steady rotations) for which the pendulum rotates about its vertical symmetry axis. Special attention is given to problems of stability and stabilization of these solutions. An analysis of the constraint reaction is performed, and parameter regions are identified in which a stabilization of the spherical robot is possible without it losing contact with the plane. It is shown that the partial solutions can be stabilized by varying the angular velocity of rotation of the pendulum about its symmetry axis, and that the rotation of the pendulum is a necessary condition for stabilization without the robot losing contact with the plane.

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. 329352
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 wellknown 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 threedimensional Poincaré sections shows that limit cycles can arise in this case in the neighborhood of the saddle point. 
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. 212233
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.

Kilin A. A., Artemova E. M.
Integrability and Chaos in Vortex Lattice Dynamics
2019, vol. 24, no. 1, pp. 101113
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.

Kilin A. A., Pivovarova E. N.
Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges
2018, vol. 23, no. 78, pp. 887907
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
righthand 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.

Borisov A. V., Kilin A. A., Mamaev I. S.
A Nonholonomic Model of the Paul Trap
2018, vol. 23, no. 3, pp. 339354
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 threedimensional 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.

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. 298317
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 righthand 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.

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. 78, pp. 874884
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 subRiemannian geodesics on $SE(3)$ are obtained.

Karavaev Y. L., Kilin A. A., Klekovkin A. V.
Experimental Investigations of the Controlled Motion of a Screwless Underwater Robot
2016, vol. 21, no. 78, pp. 918926
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.

Klenov A. I., Kilin A. A.
Influence of Vortex Structures on the Controlled Motion of an Abovewater Screwless Robot
2016, vol. 21, no. 78, pp. 927938
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.

Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.
Integrability and Nonintegrability of SubRiemannian Geodesic Flows on Carnot Groups
2016, vol. 21, no. 6, pp. 759774
Abstract
This paper is concerned with two systems from subRiemannian 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.

Kilin A. A., Pivovarova E. N., Ivanova T. B.
Spherical Robot of Combined Type: Dynamics and Control
2015, vol. 20, no. 6, pp. 716728
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.

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. 739751
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.

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. 752766
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 quasivelocities. 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 onewheeled vehicle and a wheeled vehicle with two rotating wheel pairs.

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. 134152
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 steadystate 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.

Borisov A. V., Kilin A. A., Mamaev I. S.
Dynamics and Control of an Omniwheel Vehicle
2015, vol. 20, no. 2, pp. 153172
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.

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. 832859
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.

Borisov A. V., Kilin A. A., Mamaev I. S.
The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem
2013, vol. 18, no. 12, pp. 3362
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 pseudoleapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.

Borisov A. V., Kilin A. A., Mamaev I. S.
How to Control the Chaplygin Ball Using Rotors. II
2013, vol. 18, no. 12, pp. 144158
Abstract
In our earlier paper [3] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with noslip 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 dissipationfree periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.

Borisov A. V., Kilin A. A., Mamaev I. S.
How to Control Chaplygin’s Sphere Using Rotors
2012, vol. 17, no. 34, pp. 258272
Abstract
In the paper we study the control of a balanced dynamically nonsymmetric sphere with rotors. The noslip 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.

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. 170190
Abstract
We discuss explicit integration and bifurcation analysis of two nonholonomic problems. One of them is the Chaplygin’s problem on noslip rolling of a balanced dynamically nonsymmetric 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 nontransparent 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 sphericalsupport system does not affect its integrability.

Borisov A. V., Kilin A. A., Mamaev I. S.
On the Model of Nonholonomic Billiard
2011, vol. 16, no. 6, pp. 653662
Abstract
In this paper we develop a new model of nonholonomic 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 threedimensionalpointmap technique, the nonintegrability of the nonholonomic billiard within an ellipse is shown.

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. 465483
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 nonholonomic 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.

Borisov A. V., Kilin A. A., Mamaev I. S.
Hamiltonicity and integrability of the Suslov problem
2011, vol. 16, no. 12, pp. 104116
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.

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. 615620
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 onedimensional $N$particle system discussed in the recent paper [2] and show that for the latter system an analogous superintegral can be constructed.

Borisov A. V., Mamaev I. S., Kilin A. A.
The Hamiltonian Dynamics of Selfgravitating Liquid and Gas Ellipsoids
2009, vol. 14, no. 2, pp. 179217
Abstract
The dynamics of selfgravitating 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).

Borisov A. V., Kilin A. A., Mamaev I. S.
Multiparticle Systems. The Algebra of Integrals and Integrable Cases
2009, vol. 14, no. 1, pp. 1841
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. 
Borisov A. V., Kilin A. A., Mamaev I. S.
Stability of Steady Rotations in the Nonholonomic Routh Problem
2008, vol. 13, no. 4, pp. 239249
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.

Borisov A. V., Kilin A. A., Mamaev I. S.
Absolute and Relative Choreographies in Rigid Body Dynamics
2008, vol. 13, no. 3, pp. 204220
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 wellknown 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).

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. 221233
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.

Borisov A. V., Mamaev I. S., Kilin A. A.
Twobody problem on a sphere. Reduction, stochasticity, periodic orbits
2004, vol. 9, no. 3, pp. 265279
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 Newtoniantype 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.

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. 101111
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.

Borisov A. V., Mamaev I. S., Kilin A. A.
Dynamics of rolling disk
2003, vol. 8, no. 2, pp. 201212
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.

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. 201219
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.

Kilin A. A.
The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis
2001, vol. 6, no. 3, pp. 291306
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.

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. 233234
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. 3338) 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.

Borisov A. V., Kilin A. A.
Stability of Thomson's Configurations of Vortices on a Sphere
2000, vol. 5, no. 2, pp. 189200
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.

Kilin A. A.
Libration points in spaces $S^2$ and $L^2$
1999, vol. 4, no. 1, pp. 91103
Abstract
We consider twobody problem and restricted threebody problem in spaces $S^2$ and $L^2$. For twobody 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.
