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    Alexey Borisov

    Alexey Borisov
    1, Universitetskaya str., Izhevsk 426034, Russia
    Institute of Computer Science, Russia

    Director, Institute of Computer Science, Udmurt State University, Russia
    Professor, Department of Computational Mechanics at UdSU
    Director, Scientific and Publishing Center "Regular and Chaotic Dynamics"

    Born: March 27, 1965 in Moscow, Russia
    1984-1989: student of N.E. Bauman Moscow State Technical University (MSTU).
    1992: Ph.D. (candidate of science). Thesis title: "Nonintegrability of Kirchhoff equations and related problems in rigid body dynamics", M.V. Lomonosov Moscow State University.
    2001: Doctor in physics and mathematics. Thesis title: "Poisson structures and Lie Algebras in Hamiltonian Mechanics", M.V. Lomonosov Moscow State University.

    Positions held:
    1996-2001: Head of the Laboratory of Dynamical Chaos and Nonlinearity at the Udmurt State University, Izhevsk.
    since 1998: Director of the Scientific and Publishing Center "Regular and Chaotic Dynamics".
    since 2002: Head of the Laboratory of Nonlinear Dynamics at A.A. Blagonravov Mechanical Engineering Research Institute of Russian Academy of Sciences, Moscow.
    since 2002: Head of the Department of the Mathematical Methods in Nonlinear Dynamics at the Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences.
    since 2010: Vice-rector for information and computer technology of UdSU

    Member of the Russian National Committee on Theoretical and Applied Mechanics (2001)
    Corresponding Member of Russian Academy of Natural Sciences (2006)

    Сo-founder and associate editor of the International Scientific Journal "Regular and Chaotic Dynamics"; co-founder and editor-in-chief of "Nelineinaya Dinamika" (Russian Journal of Nonlinear Dynamics).

    In 2012 A.V.Borisov and I.S.Mamaev received the Sofia Kovalevskaya Award for a series of monographs devoted to the integrable systems of Hamiltonian mechanics.

    Research supervision of 8 candidates of science and 3 doctors of science (I.S. Mamaev, A.A. Kilin, S.M. Ramodanov).


    Borisov A. V., Mamaev I. S.
    An inhomogeneous Chaplygin sleigh
    2017, Vol. 13, No. 4, pp.  625–639
    In this paper we investigate the dynamics of a system that is a generalization of the Chaplygin sleigh to the case of an inhomogeneous nonholonomic constraint. We perform an explicit integration and a sufficiently complete qualitative analysis of the dynamics.
    Keywords: Chaplygin sleigh, inhomogeneous nonholonomic constraints, conservation laws, qualitative analysis, resonance
    Citation: Borisov A. V., Mamaev I. S.,  An inhomogeneous Chaplygin sleigh, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 4, pp. 625–639
    Bizyaev I. A., Borisov A. V., Mamaev I. S.
    This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.
    Keywords: invariant submanifold, rotation number, Cantor ladder, limit cycles
    Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  The Hess–Appelrot case and quantization of the rotation number, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 433-452
    Borisov A. V., Kazakov A. O., Pivovarova E. N.
    This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of perioddoubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.
    Keywords: Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact
    Citation: Borisov A. V., Kazakov A. O., Pivovarova E. N.,  Regular and chaotic dynamics in the rubber model of a Chaplygin top, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 2, pp. 277-297
    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
    Bizyaev I. A., Borisov A. V., Mamaev I. S.
    Dynamics of the Chaplygin sleigh on a cylinder
    2016, Vol. 12, No. 4, pp.  675–687
    This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.
    Keywords: Chaplygin sleigh, invariant measure, nonholonomic mechanics
    Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  Dynamics of the Chaplygin sleigh on a cylinder, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 675–687
    Borisov A. V., Mamaev I. S., Bizyaev I. A.
    In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.
    Keywords: nonholonomic mechanics, nonholonomic constraint, d’Alembert–Lagrange principle, permutation relations
    Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A.,  Historical and critical review of the development of nonholonomic mechanics: the classical period, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 3, pp. 385-411
    Bizyaev I. A., Borisov A. V., Kazakov A. O.
    In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems.We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.
    Keywords: Suslov problem, nonholonomic constraint, reversal, strange attractor
    Citation: Bizyaev I. A., Borisov A. V., Kazakov A. O.,  Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 2, pp. 263-287
    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
    Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S.
    Topology and Bifurcations in Nonholonomic Mechanics
    2015, Vol. 11, No. 4, pp.  735–762
    This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie –Poisson bracket of rank 2. This Lie – Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.
    Keywords: nonholonomic hinge, topology, bifurcation diagram, tensor invariants, Poisson bracket, stability
    Citation: Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S.,  Topology and Bifurcations in Nonholonomic Mechanics, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 4, pp. 735–762
    Borisov A. V., Mamaev I. S.
    Symmetries and Reduction in Nonholonomic Mechanics
    2015, Vol. 11, No. 4, pp.  763–823
    This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.
    Keywords: reduction, symmetry, tensor invariant, first integral, symmetry group, symmetry field, nonholonomic constraint, Noether theorem
    Citation: Borisov A. V., Mamaev I. S.,  Symmetries and Reduction in Nonholonomic Mechanics, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 4, pp. 763–823
    Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V.
    In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
    Keywords: dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
    Citation: Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V.,  On the dynamics of a body with an axisymmetric base sliding on a rough plane, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 3, pp. 547-577
    Borisov A. V., Mamaev I. S., Bizyaev I. A.
    The Jacobi Integral in NonholonomicMechanics
    2015, Vol. 11, No. 2, pp.  377-396
    In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.
    Keywords: nonholonomic constraint, Jacobi integral, Chaplygin sleigh, rotating table, Suslov problem
    Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A.,  The Jacobi Integral in NonholonomicMechanics, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 2, pp. 377-396
    Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S.
    In this paper we investigate the dynamics of a body with a flat base sliding on a inclined plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. Computer-aided analysis of the system’s dynamics on the inclined plane using phase portraits has allowed us to reveal dynamical effects that have not been found earlier.
    Keywords: dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
    Citation: Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S.,  On the dynamics of a body with an axisymmetric base sliding on a rough plane, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 483-495
    Bizyaev I. A., Borisov A. V., Mamaev I. S.
    The dynamics of three vortex sources
    2014, Vol. 10, No. 3, pp.  319-327
    In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system’s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.
    Keywords: integrability, vortex sources, shape sphere, reduction, homothetic configurations
    Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  The dynamics of three vortex sources, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 3, pp. 319-327
    Borisov A. V., Mamaev I. S.
    This paper discusses new unresolved problems of nonholonomic mechanics. Hypotheses of the possibility of Hamiltonization and the existence of an invariant measure for such systems are advanced.
    Keywords: nonholonomic mechanics, tensor invariant, invariant measure, Poisson structure
    Citation: Borisov A. V., Mamaev I. S.,  Invariant Measure and Hamiltonization of Nonholonomic Systems, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 3, pp. 355-359
    Borisov A. V., Kazakov A. O., Sataev I. R.
    We study both analytically and numerically the dynamics of an inhomogeneous ball on a rough horizontal plane under the infuence of gravity. A nonholonomic constraint of zero velocity at the point of contact of the ball with the plane is imposed. In the case of an arbitrary displacement of the center of mass of the ball, the system is nonintegrable without the property of phase volume conservation. We show that at certain parameter values the unbalanced ball exhibits the effect of reversal (the direction of the ball rotation reverses). Charts of dynamical regimes on the parameter plane are presented. The system under consideration exhibits diverse chaotic dynamics, in particular, the figure-eight chaotic attractor, which is a special type of pseudohyperbolic chaos.
    Keywords: Chaplygin’s top, rolling without slipping, reversibility, involution, integrability, reverse, chart of dynamical regimes, strange attractor
    Citation: Borisov A. V., Kazakov A. O., Sataev I. R.,  Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 3, pp. 361-380
    Bizyaev I. A., Borisov A. V., Mamaev I. S.
    This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with density stratification and a steady-state velocity field. As in the classical setting, it is assumed that the figure or its layers uniformly rotate about an axis fixed in space. As is well known, when there is no rotation, only a ball can be a figure of equilibrium.

    It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with inherent constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification.

    We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.
    Keywords: self-gravitating fluid, confocal stratification, homothetic stratification, space of constant curvature
    Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  Figures of equilibrium of an inhomogeneous self-gravitating fluid, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 1, pp. 73-100
    Bolsinov A. V., Borisov A. V., Mamaev I. S.
    This paper develops the theory of the reducing multiplier for a special class of nonholonomic dynamical systems, when the resulting nonlinear Poisson structure is reduced to the Lie–Poisson bracket of the algebra $e(3)$. As an illustration, the Chaplygin ball rolling problem and the Veselova system are considered. In addition, an integrable gyrostatic generalization of the Veselova system is obtained.
    Keywords: nonholonomic dynamical system, Poisson bracket, Poisson structure, reducing multiplier, Hamiltonization, conformally Hamiltonian system, Chaplygin ball
    Citation: Bolsinov A. V., Borisov A. V., Mamaev I. S.,  Geometrization of the Chaplygin reducing-multiplier theorem, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 627-640
    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
    Borisov A. V., Mamaev I. S., Karavaev Y. L.
    On the loss of contact of the Euler disk
    2013, Vol. 9, No. 3, pp.  499-506
    The paper presents experimental investigation of a homogeneous circular disk rolling on a horizontal plane. In this paper two methods of experimental determination of the loss of contact between the rolling disk and the horizontal surface before the abrupt halt are proposed. Experimental results for disks of different masses and different materials are presented. The reasons for “micro losses” of contact with surface revealed during the rolling are discussed.
    Keywords: Euler disk, loss of contact, experiment
    Citation: Borisov A. V., Mamaev I. S., Karavaev Y. L.,  On the loss of contact of the Euler disk, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 3, pp. 499-506
    Bizyaev I. A., Borisov A. V., Mamaev I. S.
    In this paper we investigate two systems consisting of a spherical shell rolling on a plane without slipping and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is fixed at the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of the nonholonomic hinge. The equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics.
    Keywords: nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge
    Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 3, pp. 547-566
    Borisov A. V., Mamaev I. S., Bizyaev I. A.
    In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.
    Keywords: nonholonomic constraint, tensor invariant, first integral, invariant measure, integrability, conformally Hamiltonian system, rubber rolling, reversible, involution
    Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A.,  The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 2, pp. 141-202
    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., Mamaev I. S.
    A new integrable system describing the rolling of a rigid body with a spherical cavity over a spherical base is considered. Previously the authors found the separation of variables for this system at the zero level of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables.
    Keywords: integrable system, bifurcation diagram, conformally Hamiltonian system, bifurcation, Liouville foliation, critical periodic solution
    Citation: Borisov A. V., Mamaev I. S.,  Topological analysis of one integrable system related to the rolling of a ball over a sphere, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 5, pp. 957-975
    Borisov A. V., Mamaev I. S., Treschev D. V.
    In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.
    Keywords: rolling without slipping, nonholonomic constraint, Chaplygin system, conformally Hamiltonian system
    Citation: Borisov A. V., Mamaev I. S., Treschev D. V.,  Rolling of a rigid body without slipping and spinning: kinematics and dynamics, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 4, pp. 783-797
    Bolsinov A. V., Borisov A. V., Mamaev I. S.
    In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.
    Keywords: non-holonomic constraint, Liouville foliation, invariant torus, invariant measure, integrability
    Citation: Bolsinov A. V., Borisov A. V., Mamaev I. S.,  Rolling without spinning of a ball on a plane: absence of an invariant measure in a system with a complete set of first integrals, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 605-616
    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., Mamaev I. S.
    We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.
    Keywords: vortex motion, non-holonomic constraint, Chaplygin ball, invariant measure, integrability, rigid body, ideal fluid
    Citation: Borisov A. V., Mamaev I. S.,  The dynamics of the Chaplygin ball with a fluid-filled cavity, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 103-111
    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
    Grinchenko V. T., Krasnopolskaya T. S., Borisov A. V., van Heijst G. J.
    Citation: Grinchenko V. T., Krasnopolskaya T. S., Borisov A. V., van Heijst G. J.,  Viatcheslav Vladimirovich Meleshko (07.10.1951–14.11.2011), Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 179-182
    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., Mamaev I. S.
    The paper considers two new integrable systems due to Chaplygin, which describe the rolling of a spherical shell on a plane, with a ball or Lagrange’s gyroscope inside. All necessary first integrals and an invariant measure are found. The reduction to quadratures is given.
    Keywords: non-holonomic constraint, integrability, invariant measure, gyroscope, quadrature, coupled rigid bodies
    Citation: Borisov A. V., Mamaev I. S.,  Two non-holonomic integrable systems of coupled rigid bodies, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 559-568
    Bolsinov A. V., Borisov A. V., Mamaev I. S.
    The paper is concerned with the use of bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We give the proof of the theorem on the appearance (disappearance) of fixed points in the case of the Morse index change. New relative equilibria in the problem of the motion of point vortices of equal intensity in a circle are found.
    Keywords: Morse index, Conley index, bifurcation analysis, bifurcation diagram, Hamiltonian dynamics, fixed point, relative equilibrium
    Citation: Bolsinov A. V., Borisov A. V., Mamaev I. S.,  The bifurcation analysis and the Conley index in mechanics, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 649-681
    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., Gazizullina L., Mamaev I. S.
    On V.A. Steklov’s legacy in classical mechanics
    2011, Vol. 7, No. 2, pp.  389-403
    This paper has been written for a collection of V.A. Steklov’s selected works, which is being prepared for publication and is entitled «Works on Mechanics 1902–1909: Translations from French». The collection is based on V.A. Steklov’s papers on mechanics published in French journals from 1902 to 1909.
    Citation: Borisov A. V., Gazizullina L., Mamaev I. S.,  On V.A. Steklov’s legacy in classical mechanics, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 2, pp. 389-403
    Borisov A. V., Mamaev I. S., Vaskina A. V.
    This paper presents a topological approach to the search and stability analysis of relative equilibria of three point vortices of equal intensities. It is shown that the equations of motion can be reduced by one degree of freedom. We have found two new stationary configurations (isosceles and non-symmetrical collinear) and studied their bifurcations and stability.
    Keywords: point vortex, reduction, bifurcational diagram, relative equilibriums, stability, periodic solutions
    Citation: Borisov A. V., Mamaev I. S., Vaskina A. V.,  Stability of new relative equilibria of the system of three point vortices in a circular domain, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 119-138
    Borisov A. V., Mamaev I. S., Ivanova T. B.
    We consider figures of equilibrium and stability of a liquid self-gravitating elliptic cylinder. The flow within the cylinder is assumed to be dew to an elliptic perturbation. A bifurcation diagram is plotted and conditions for steady solutions to exist are indicated.
    Keywords: self-gravitating liquid, elliptic cylinder, bifurcation point, stability, Riemann equations
    Citation: Borisov A. V., Mamaev I. S., Ivanova T. B.,  Stability of a liquid self-gravitating elliptic cylinder with intrinsic rotation, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 4, pp. 807-822
    Bolsinov A. V., Borisov A. V., Mamaev I. S.
    Hamiltonisation problem for non-holonomic systems, both integrable and non-integrable, is considered. This question is important for qualitative analysis of such systems and allows one to determine possible dynamical effects. The first part is devoted to the representation of integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighbourhood of a periodic solution is proved for an arbitrary measure preserving system (including integrable). General consructions are always illustrated by examples from non-holonomic mechanics.
    Keywords: conformally Hamiltonian system, nonholonomic system, invariant measure, periodic trajectory, invariant torus, integrable system
    Citation: Bolsinov A. V., Borisov A. V., Mamaev I. S.,  Hamiltonisation of non-holonomic systems in the neighborhood of invariant manifolds, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 4, pp. 829-854
    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., Mamaev I. S.
    Reply to A. T. Fomenko's comments
    2010, Vol. 6, No. 4, pp.  893-895
    Citation: Borisov A. V., Mamaev I. S.,  Reply to A. T. Fomenko's comments, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 4, pp. 893-895
    Borisov A. V.
    Reply to V.F. Zhuravlev
    2010, Vol. 6, No. 4, pp.  897-901
    Citation: Borisov A. V.,  Reply to V.F. Zhuravlev, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 4, pp. 897-901
    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., Mamaev I. S., Ramodanov S. M.
    Dynamic advection
    2010, Vol. 6, No. 3, pp.  521-530
    A new concept of dynamic advection is introduced. The model of dynamic advection deals with the motion of massive particles in a 2D flow of an ideal incompressible liquid. Unlike the standard advection problem, which is widely treated in the modern literature, our equations of motion account not only for particles’ kinematics, governed by the Euler equations, but also for their dynamics (which is obviously neglected if the mass of particles is taken to be zero). A few simple model problems are considered.
    Keywords: advection, mixing, point vortex, coarse-grained impurities, bifurcation complex
    Citation: Borisov A. V., Mamaev I. S., Ramodanov S. M.,  Dynamic advection, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 3, pp. 521-530
    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., Mamaev I. S., Ramodanov S. M.
    The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on $S^2$ are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.
    Keywords: hydrodynamics on a sphere, coupled body-vortex system, mass vortex, equations of motion, integrability
    Citation: Borisov A. V., Mamaev I. S., Ramodanov S. M.,  Coupled motion of a rigid body and point vortices on a sphere, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 3, pp. 319-343
    Borisov A. V., Mamaev I. S.
    Isomorphisms of geodesic flows on quadrics
    2009, Vol. 5, No. 2, pp.  145-158
    We consider several well-known isomorphisms between Jacobi’s geodesic problem and some integrable cases from rigid body dynamics (the cases of Clebsch and Brun). A relationship between these isomorphisms is indicated. The problem of compactification for geodesic flows on noncompact surfaces is stated. This problem is hypothesized to be intimately connected with the property of integrability.
    Keywords: quadric, geodesic flows, integrability, compactification, regularization, isomorphism
    Citation: Borisov A. V., Mamaev I. S.,  Isomorphisms of geodesic flows on quadrics, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 2, pp. 145-158
    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
    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., Mamaev I. S., Ramodanov S. M.
    The paper develops further the algebraic-reduction method for $SO(4)$-symmetric systems on the three-dimensional sphere. Canonical variables for the reduced system are constructed both on two-dimensional and three-dimensional spheres. The method is illustrated by applying it to the two-body problem on a sphere (the bodies are assumed to interact with a potential that depends only on the geodesic distance between them) and the three-vortex problem on a two-dimensional sphere.
    Keywords: Poisson structure, Lie algebra, subalgebra, Andoyer variables
    Citation: Borisov A. V., Mamaev I. S., Ramodanov S. M.,  Algebraic reduction of systems on two- and three-dimensional spheres, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 407-416
    Borisov A. V., Gazizullina L., Ramodanov S. M.
    Citation: Borisov A. V., Gazizullina L., Ramodanov S. M.,  E. Zermelo Habilitationsschrift on the vortex hydrodynamics on a sphere, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 497-513
    Borisov A. V., Mamaev I. S.
    This paper can be regarded as a continuation of our previous work [70,71] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.
    Keywords: nonholonomic systems, implementation of constraints, conservation laws, hierarchy of dynamics, explicit integration
    Citation: Borisov A. V., Mamaev I. S.,  Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 3, pp. 223-280
    Borisov A. V., Mamaev I. S., Ramodanov S. M.
    The paper deals with the derivation of the equations of motion for two spheres in an unbounded volume of ideal and incompressible fluid in 3D Euclidean space. Reduction of order, based on the use of new variables that form a Lie algebra, is offered. A trivial case of integrability is indicated.
    Keywords: motion of two spheres, ideal fluid, reduction, integrability
    Citation: Borisov A. V., Mamaev I. S., Ramodanov S. M.,  Motion of two spheres in ideal fluid. I. Equations o motions in the Euclidean space. First integrals and reduction, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 4, pp. 411-422
    Borisov A. V., Kozlov V. V., Mamaev I. S.
    We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.
    Keywords: nonholonomic mechanics, rigid body, ideal fluid, resisting medium
    Citation: Borisov A. V., Kozlov V. V., Mamaev I. S.,  Asymptotic stability and associated problems of dynamics of falling rigid body, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 3, pp. 255-296
    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., Mamaev I. S.
    We consider trajectory isomorphisms between various integrable systems on an $n$-dimensional sphere $S^n$ and a Euclidean space $R^n$. Some of the systems are classical integrable problems of Celestial Mechanics in plane and curved spaces. All the systems under consideration have an additional first integral quadratic in momentum and can be integrated analytically by using the separation of variables. We show that some integrable problems in constant curvature spaces are not essentially new from the viewpoint of the theory of integration, and they can be analyzed using known results of classical Celestial Mechanics.
    Keywords: integrable systems, two-center problem, isomorphisms
    Citation: Borisov A. V., Mamaev I. S.,  On isomorphisms of some integrable systems on a plane and a sphere, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 1, pp. 49-56
    Borisov A. V., Mamaev I. S.
    Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, Koiller and Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.
    Keywords: Chaplygin ball, rolling model, Hamiltonian structure
    Citation: Borisov A. V., Mamaev I. S.,  Rolling of a heterotgeneous ball over a sphere without sliding and spinning, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 4, pp. 445-452
    Borisov A. V., Mamaev I. S.
    We present a reduction-of-order procedure in the problem of motion of two bodies on the Lobatchevsky plane $H^2$. The bodies interact with a potential that depends only on the distance between the bodies (this holds for an analog of the Newtonian potential). In earlier works, this reduction procedure was used to analyze the motion of two bodies on the sphere
    Keywords: Lobatchevsky plane, first integral, reduction-of-order procedure, potential of interaction
    Citation: Borisov A. V., Mamaev I. S.,  Reduction in the two-body problem on the Lobatchevsky plane, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 279-285
    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
    Borisov A. V., Mamaev I. S.
    Dynamics of two vortex rings on a sphere
    2006, Vol. 2, No. 2, pp.  181-192
    The motion of two vortex rings on a sphere is considered. This motion generalizes the well-known centrally symmetrical solution of the equations of point vortex dynamics on a plane derived by D.N. Goryachev and H. Aref. The equations of motion in this case are shown to be Liouville integrable, and an explicit reduction to a Hamiltonian system with one degree of freedom is described. Two particular cases in which the solutions are periodical are presented. Explicit quadratures are given for these solutions. Phase portraits are described and bifurcation diagrams are shown for centrally symmetrical motion of four vortices on a sphere.
    Keywords: vortex, Hamiltonian, motion on a sphere, phase portrait
    Citation: Borisov A. V., Mamaev I. S.,  Dynamics of two vortex rings on a sphere, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 2, pp. 181-192
    Borisov A. V., Mamaev I. S.
    We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex by the variable separation method is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found. A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior.
    Keywords: Kirchhoff vortices, integrability, Hamiltonian, stability, point vortex
    Citation: Borisov A. V., Mamaev I. S.,  Interaction between Kirchhoff vortices and point vortices in an ideal fluid, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 2, pp. 199-213
    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
    Borisov A. V., Mamaev I. S., Ramodanov S. M.
    In this paper we consider the system of two 2D rigid circular cylinders immersed in an unbounded volume of inviscid perfect fluid. The circulations around the cylinders are assumed to be equal in magnitude and opposite in sign. Special cases of this system (the cylinders move along the line through their centers and the circulation around each cylinder is zero) are considered. A similar system of two interacting spheres was originally considered in classical works of Carl and Vilhelm Bjerknes, G. Lamb and N.E. Joukowski.

    By making the radii of the cylinders infinitesimally small, we have obtained a new mechanical system which consists of two regular point vortices but with non-zero masses. The study of this system can be reduced to the study of the motion of a particle subject to potential and gyroscopic forces. A new integrable case is found. The Hamiltonian equations of motion for this system have been generalized to the case of an arbitrary number of mass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations of motion for n point vortices.
    Keywords: perfect fluid, circulation, rigid body, qualitative analysis
    Citation: Borisov A. V., Mamaev I. S., Ramodanov S. M.,  Interaction of two circular cylinders in a perfect fluid, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp. 3-21

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