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    Alexander Kilin

    Alexander Kilin
    1, Universitetskaya str., Izhevsk 426034, Russia
    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)


    Karavaev Y. L., Klekovkin A. V., Kilin A. A.
    In this paper the model of rolling of spherical bodies on a plane without slipping is presented taking into account viscous rolling friction. Results of experiments aimed at investigating the influence of friction on the dynamics of rolling motion are presented. The proposed dynamical friction model for spherical bodies is verified and the limits of its applicability are estimated. A method for determining friction coefficients from experimental data is formulated.
    Keywords: rolling friction, dynamical model, spherical body, nonholonomic model, experimental investigation
    Citation: Karavaev Y. L., Klekovkin A. V., Kilin A. A.,  The dynamical model of the rolling friction of spherical bodies on a plane without slipping, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 4, pp.  599–609
    Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.
    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, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 1, pp.  129-146
    Vetchanin  E. V., Kilin A. A.
    This paper is concerned with the motion of an unbalanced heavy three-axial ellipsoid in an ideal fluid controlled by rotation of three internal rotors. It is proved that the motion of the body considered is controlled with respect to configuration variables except for some special cases. An explicit control that makes it possible to implement unbounded motion in an arbitrary direction has been calculated. Directions for which control actions are bounded functions of time have been determined.
    Keywords: ideal fluid, motion of a rigid body, Kirchhoff equations, control by rotors, gaits
    Citation: Vetchanin  E. V., Kilin A. A.,  Control of the motion of an unbalanced heavy ellipsoid in an ideal fluid using rotors, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp.  663–674
    Borisov A. V., Kilin A. A., Mamaev I. S.
    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, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp.  145-163
    Kilin A. A., Vetchanin  E. V.
    In this paper we consider the problem of motion of a rigid body in an ideal fluid with two material points moving along circular trajectories. The controllability of this system on the zero level set of first integrals is shown. Elementary “gaits” are presented which allow the realization of the body’s motion from one point to another. The existence of obstacles to a controlled motion of the body along an arbitrary trajectory is pointed out.
    Keywords: ideal fluid, Kirchhoff equations, controllability of gaits
    Citation: Kilin A. A., Vetchanin  E. V.,  The contol of the motion through an ideal fluid of a rigid body by means of two moving masses, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 4, pp.  633–645
    Kilin A. A., Karavaev Y. L.
    This paper presents the results of experimental investigations for the rolling of a spherical robot of combined type actuated by an internal wheeled vehicle with rotor on a horizontal plane. The control of spherical robot based on nonholonomic dynamical by means of gaits. We consider the motion of the spherical robot in case of constant control actions, as well as impulse control. A number of experiments have been carried out confirming the importance of rolling friction.
    Keywords: spherical robot of combined type, dynamic model, control by means of gaits, rolling friction
    Citation: Kilin A. A., Karavaev Y. L.,  Experimental research of dynamic of spherical robot of combined type, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 4, pp.  721–734
    Karavaev Y. L., Kilin A. A.
    The dynamic model for a spherical robot with an internal omniwheel platform is presented. Equations of motion and first integrals according to the non-holonomic model are given. We consider particular solutions and their stability. The algorithm of control of spherical robot for movement along a given trajectory are presented.
    Keywords: spherical robot, dynamical model, non-holonomic constraint, omniwheel, stability
    Citation: Karavaev Y. L., Kilin A. A.,  The dynamic of a spherical robot with an internal omniwheel platform, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 1, pp.  187-204
    Kilin A. A., Bobykin A. D.
    Control of a Vehicle with Omniwheels on a Plane
    2014, Vol. 10, No. 4, pp.  473-481
    The problem of motion of a vehicle in the form of a platform with an arbitrary number of Mecanum wheels fastened on it is considered. The controllability of this vehicle is discussed within the framework of the nonholonomic rolling model. An explicit algorithm is presented for calculating the control torques of the motors required to follow an arbitrary trajectory. Examples of controls for executing the simplest maneuvers are given.
    Keywords: omniwheel, roller bearing wheel, nonholonomic constraint, dynamical system, integrability, controllability
    Citation: Kilin A. A., Bobykin A. D.,  Control of a Vehicle with Omniwheels on a Plane, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp.  473-481
    Kilin A. A., Karavaev Y. L.
    The kinematic control model for a spherical robot with an internal omniwheel platform is presented. We consider singularities of control of spherical robot with an unbalanced internal omniwheel platform. The general algorithm of control of spherical robot according to the kinematical quasi-static model and controls for simple trajectories (a straight line and in a circle) are presented. Experimental investigations have been carried out for all introduced control algorithms.
    Keywords: spherical robot, kinematic model, nonholonomic constraint, omniwheel, displacement of center of mass
    Citation: Kilin A. A., Karavaev Y. L.,  The kinematic control model for a spherical robot with an unbalanced internal omniwheel platform, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp.  497-511
    Kilin A. A., Karavaev Y. L., Klekovkin A. V.
    In this article a kinematic model of the spherical robot is considered, which is set in motion by the internal platform with omni-wheels. It has been introduced a description of construction, algorithm of trajectory planning according to developed kinematic model, it has been realized experimental research for typical trajectories: moving along a straight line and moving along a circle.
    Keywords: spherorobot, kinematic model, non-holonomic constraint, omni-wheel
    Citation: Kilin A. A., Karavaev Y. L., Klekovkin A. V.,  Kinematic control of a high manoeuvrable mobile spherical robot with internal omni-wheeled platform, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 1, pp.  113-126
    Borisov A. V., Kilin A. A., Mamaev I. S.
    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 a 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, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp.  721-754
    Bolsinov A. V., Kilin A. A., Kazakov A. O.
    Topological monodromy in nonholonomic systems
    2013, Vol. 9, No. 2, pp.  203-227
    The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.
    Keywords: topological monodromy, integrable systems, nonholonomic systems, Poincaré map, bifurcation analysis, focus-focus singularities
    Citation: Bolsinov A. V., Kilin A. A., Kazakov A. O.,  Topological monodromy in nonholonomic systems, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 2, pp.  203-227
    Borisov A. V., Kilin A. A., Mamaev I. S.
    How to control the Chaplygin ball using rotors. II
    2013, Vol. 9, No. 1, pp.  59-76
    In our earlier paper [2] 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, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 1, pp.  59-76
    Borisov A. V., Kilin A. A., Mamaev I. S.
    How to control the Chaplygin sphere using rotors
    2012, Vol. 8, No. 2, pp.  289-307
    In the paper we study control of a balanced dynamically nonsymmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic contrability is shown and the control inputs providing motion of 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 the Chaplygin sphere using rotors, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 2, pp.  289-307
    Borisov A. V., Kilin A. A., Mamaev I. S.
    We consider the problem of the 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 in which the mutual 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, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp.  113-147
    Borisov A. V., Kilin A. A., Mamaev I. S.
    An omni-wheel vehicle on a plane and a sphere
    2011, Vol. 7, No. 4, pp.  785-801
    We consider a nonholonomic model of the dynamics of an omni-wheel vehicle on a plane and a sphere. An elementary derivation of equations is presented, the dynamics of a free system is investigated, a relation to control problems is shown.
    Keywords: omni-wheel, roller-bearing wheel, nonholonomic constraint, dynamical system, invariant measure, integrability, controllability
    Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  An omni-wheel vehicle on a plane and a sphere, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 4, pp.  785-801
    Borisov A. V., Kilin A. A., Mamaev I. S.
    We consider the problem of explicit integration and bifurcation analysis for two systems of nonholonomic mechanics. The first one is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetrical ball on a horizontal plane. The second problem is on the motion of rigid body in a spherical support. We explicitly integrate this problem by generalizing the transformation which Chaplygin applied to the integration of the problem of the rolling ball at a non-zero constant of areas. We consider the geometric interpretation of this transformation from the viewpoint of a trajectory isomorphism between two systems at different levels of the energy integral. Generalization of this transformation for the case of dynamics in a spherical support allows us to integrate the equations of motion explicitly in quadratures and, in addition, to indicate periodic solutions and analyze their stability. We also show that adding a gyrostat does not lead to the loss of 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, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 2, pp.  313-338
    Borisov A. V., Kilin A. A., Mamaev I. S.
    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, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 4, pp.  869-889
    Borisov A. V., Bolotin S. V., Kilin A. A., Mamaev I. S., Treschev D. V.
    Valery Vasilievich Kozlov. On his 60th birthday
    2010, Vol. 6, No. 3, pp.  461-488
    Citation: Borisov A. V., Bolotin S. V., Kilin A. A., Mamaev I. S., Treschev D. V.,  Valery Vasilievich Kozlov. On his 60th birthday, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 3, pp.  461-488
    Borisov A. V., Kilin A. A., Mamaev I. S.
    On the model of non-holonomic billiard
    2010, Vol. 6, No. 2, pp.  373-385
    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 mapping, 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, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp.  373-385
    Borisov A. V., Kilin A. A., Mamaev I. S.
    We consider the problems of Hamiltonian representation and integrability of the nonholonomic Suslov system and its generalization suggested by S. A. Chaplygin. These aspects are very important for understanding the dynamics and qualitative analysis of the system. In particular, they are related to the nontrivial asymptotic behaviour (i. e. to some scattering problem). The paper presents a general approach based on the study of the hierarchy of dynamical behaviour of nonholonomic systems.
    Keywords: Hamiltonian system, Poisson bracket, nonholonomic constraint, invariant measure, integrability
    Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Hamiltonian representation and integrability of the Suslov problem, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 1, pp.  127-142
    Borisov A. V., Kilin A. A., Mamaev I. S.
    New superintegrable system on a sphere
    2009, Vol. 5, No. 4, pp.  455-462
    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 [3], 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 [13] 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.,  New superintegrable system on a sphere, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp.  455-462
    Borisov A. V., Kilin A. A., Mamaev I. S.
    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 particle-interaction potential homogeneous of degree $α=-2$ 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 $α=-2$ 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, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 1, pp.  53-82
    Kilin A. A.
    The Jacobi problem on a plane
    2009, Vol. 5, No. 1, pp.  83-86
    3-particle systems with a particle-interaction homogeneous potential of degree $α=-2$ is considered. A constructive procedure of reduction of the system by 2 degrees of freedom is performed. The nonintegrability of the systems is shown using the Poincare mapping.
    Keywords: multiparticle system, potential, Hamiltonian, reduction, integrability
    Citation: Kilin A. A.,  The Jacobi problem on a plane, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 1, pp.  83-86
    Borisov A. V., Mamaev I. S., Kilin A. A.
    The paper contains the review and original results on the dynamics of liquid and gas self-gravitating ellipsoids. Equations of motion are given in Lagrangian and Hamiltonian form, in particular, the Hamiltonian formalism on Lie algebras is presented. Problems of nonintegrability and chaotical behavior of the system are formulated and studied. We also classify all known integrable cases and give some hypotheses about nonintegrability in the general case. Results of numerical modelling are presented, which can be considered as a computer proof of nonintegrability.
    Keywords: liquid and gas self-gravitating ellipsoids, integrability, chaotic behavior
    Citation: Borisov A. V., Mamaev I. S., Kilin A. A.,  Hamiltonian Dynamics of Liquid and Gas Self-Gravitating Ellipsoids, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp.  363-406
    Borisov A. V., Kilin A. A., Mamaev I. S.
    The dynamics of an antipodal vortex on a sphere (a point vortex plus its antipode with opposite circulation) is considered. It is shown that the system of n antipodal vortices can be reduced by four dimensions (two degrees of freedom). The cases n=2,3 are explored in greater detail both analytically and numerically. We discuss Thomson, collinear and isosceles configurations of antipodal vortices and study their bifurcations.
    Keywords: hydrodynamics, ideal fluid, vortex dynamics, point vortex, reduction, bifurcation analysis
    Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  A New Integrable Problem of Motion of Point Vortices on the Sphere, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 2, pp.  211-223
    Borisov A. V., Kilin A. A., Mamaev I. S.
    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 in 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 non-holonomic Routh problem, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp.  333-345
    Kilin A. A.
    In the paper we present a new integral of motion in the problem of rolling motion of a heavy symmetric sphere on the surface of a paraboloid. We use this integral to study the Lyapunov stability of some trivial steady rotations.
    Keywords: dynamical sysytem, non-holonomic constraint, integral, periodic solution, Lyapunov stability
    Citation: Kilin A. A.,  New integral in nonholonomic Painleve-Chaplygin problem, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 2, pp.  193-198
    Borisov A. V., Kilin A. A., Mamaev I. S.
    The paper deals with a transition to chaos in the phase-plane portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotisation have been indicated: 1) growth of the homoclinic structure and 2) development of cascades of period doubling bifurcations. On the zero level of the integral of areas, an adiabatic behavior of the system (as the energy tends to zero) has been noticed. Meander tori induced by the breakdown of the torsion property of the mapping have been found.
    Keywords: motion of a rigid body, phase-plane portrait, mechanism of chaotisation, 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, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp.  191-207
    Borisov A. V., Kilin A. A., Mamaev I. S.
    We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincare surface-of-section technique to perform the reduction a four-vortex system on a sphere.
    Keywords: reduction, point vortex, equations of motion, Poincare map
    Citation: Borisov A. V., Kilin A. A., Mamaev I. S.,  Reduction and chaotic behavior of point vortices on a plane and a sphere, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp.  233-246
    Borisov A. V., Kilin A. A., Mamaev I. S.
    For the classical problem of motion of a rigid body about a fixed point with zero integral of areas, the paper presents a family of solutions which are periodic in the absolute space. Such solutions are known as choreographies. The family includes the famous Delaunay solution in the case of Kovalevskaya, some particular solutions in the Goryachev-Chaplygin case and Steklov’s solution. The «genealogy» of the solutions of the family, arising naturally 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 but 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, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp.  123-141

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