0
2013
Impact Factor

Alexander Kilin

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:

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

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