Alexey Borisov
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
19841989: 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:
19962001: 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: Vicerector 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)
Сofounder and associate editor of the International Scientific Journal "Regular and Chaotic Dynamics"; cofounder and editorinchief 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).
Publications:
Borisov A. V., Mikishanina E. A.
Abstract
This work is devoted to the study of the dynamics of the Chaplygin ball with variable
moments of inertia, which occur due to the motion of pairs of internal material points, and
internal rotors. The components of the inertia tensor and the gyrostatic momentum are periodic
functions. In general, the problem is nonintegrable. In a special case, the relationship of the
problem under consideration with the Liouville problem with changing parameters is shown.
The case of the Chaplygin ball moving from rest is considered separately. Poincaré maps are
constructed, strange attractors are found, and the stages of the origin of strange attractors are
shown. Also, the trajectories of contact points are constructed to confirm the chaotic dynamics
of the ball. A chart of dynamical regimes is constructed in a separate case for analyzing the
nature of strange attractors.

Borisov A. V., Mamaev I. S.
Abstract
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.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Abstract
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.

Borisov A. V., Kazakov A. O., Pivovarova E. N.
Abstract
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 quasiperiodic.

Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
This paper is concerned with two systems from subRiemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Abstract
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 (twodimensional) 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.

Borisov A. V., Mamaev I. S., Bizyaev I. A.
Abstract
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.

Bizyaev I. A., Borisov A. V., Kazakov A. O.
Abstract
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 bodyfixed 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.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasivelocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a onewheeled vehicle and a wheeled vehicle with two rotating wheel pairs.

Borisov A. V., Mamaev I. S.
Abstract
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.

Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S.
Abstract
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.

Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V.
Abstract
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 highspeed 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 wellknown experimental and theoretical results in this area.

Borisov A. V., Mamaev I. S., Bizyaev I. A.
Abstract
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.

Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S.
Abstract
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. Computeraided 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.

Borisov A. V., Kazakov A. O., Sataev I. R.
Abstract
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 figureeight chaotic attractor, which is a special type of pseudohyperbolic chaos.

Borisov A. V., Mamaev I. S.
Abstract
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.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Abstract
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.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Abstract
This paper is concerned with the figures of equilibrium of a selfgravitating ideal fluid with density stratification and a steadystate 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. 
Borisov A. V., Kilin A. A., Mamaev I. S.
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 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.

Bolsinov A. V., Borisov A. V., Mamaev I. S.
Abstract
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.

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
Abstract
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.

Borisov A. V., Mamaev I. S., Karavaev Y. L.
Abstract
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.

Borisov A. V., Mamaev I. S., Bizyaev I. A.
Abstract
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 quasiHamiltonian behavior.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
In our earlier paper [2] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with noslip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipationfree periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.

Borisov A. V., Mamaev I. S.
Abstract
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.

Borisov A. V., Mamaev I. S., Treschev D. V.
Abstract
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 rubberrolling (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.

Bolsinov A. V., Borisov A. V., Mamaev I. S.
Abstract
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 2tori; 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.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
In the paper we study control of a balanced dynamically nonsymmetric sphere with rotors. The noslip 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.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
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 pseudoleapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.
