0
2013
Impact Factor

Ivan Bizyaev

Publications:

Bizyaev I. A.
The Inertial Motion of a Roller Racer
2017, vol. 22, no. 3, pp.  239-247
Abstract
This paper addresses the problem of the inertial motion of a roller racer, which reduces to investigating a dynamical system on a (two-dimensional) torus and to classifying singular points on it. It is shown that the motion of the roller racer in absolute space is asymptotic. A restriction on the system parameters in which this motion is bounded (compact) is presented.
Keywords: roller racer, invariant measure, nonholonomic mechanics, scattering map
Citation: Bizyaev I. A.,  The Inertial Motion of a Roller Racer, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 239-247
DOI:10.1134/S1560354717030042
Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Hess–Appelrot Case and Quantization of the Rotation Number
2017, vol. 22, no. 2, pp.  180-196
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.
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, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 180-196
DOI:10.1134/S156035471702006X
Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.
Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups
2016, vol. 21, no. 6, pp.  759-774
Abstract
This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector $(3, 6, 14)$, the other is defined by two generatrices and growth vector $(2, 3, 5, 8)$. Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.
Keywords: sub-Riemannian geometry, Carnot group, Poincaré map, first integrals
Citation: Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.,  Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 759-774
DOI:10.1134/S1560354716060125
Borisov A. V., Mamaev I. S., Bizyaev I. A.
The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity
2016, vol. 21, no. 5, pp.  556-580
Abstract
In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature $S^2$ and $S^3$. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincaré section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems.
Keywords: celestial mechanics, space of constant curvature, reduction, rigid body dynamics, Poincaré section
Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A.,  The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 556-580
DOI:10.1134/S1560354716050075
Borisov A. V., Mamaev I. S., Bizyaev I. A.
Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period
2016, vol. 21, no. 4, pp.  455-476
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.
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, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 455-476
DOI:10.1134/S1560354716040055
Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Dynamics of Vortex Sources in a Deformation Flow
2016, vol. 21, no. 3, pp.  367-376
Abstract
This paper is concerned with the dynamics of vortex sources in a deformation flow. The case of two vortex sources is shown to be integrable by quadratures. In addition, the relative equilibria (of the reduced system) are examined in detail and it is shown that in this case the trajectory of vortex sources is an ellipse.
Keywords: integrability, vortex sources, reduction, deformation flow
Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  The Dynamics of Vortex Sources in a Deformation Flow, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 367-376
DOI:10.1134/S1560354716030084
Bizyaev I. A., Borisov A. V., Mamaev I. S.
Dynamics of the Chaplygin Sleigh on a Cylinder
2016, vol. 21, no. 1, pp.  136-146
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 (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, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 136-146
DOI:10.1134/S1560354716010081
Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A.
Qualitative Analysis of the Dynamics of a Wheeled Vehicle
2015, vol. 20, no. 6, pp.  739-751
Abstract
This paper is concerned with the problem of the motion of a wheeled vehicle on a plane in the case where one of the wheel pairs is fixed. In addition, the motion of a wheeled vehicle on a plane in the case of two free wheel pairs is considered. A method for obtaining equations of motion for the vehicle with an arbitrary geometry is presented. Possible kinds of motion of the vehicle with a fixed wheel pair are determined.
Keywords: nonholonomic constraint, system dynamics, wheeled vehicle, Chaplygin system
Citation: Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A.,  Qualitative Analysis of the Dynamics of a Wheeled Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 739-751
DOI:10.1134/S156035471506009X
Bizyaev I. A., Borisov A. V., Kazakov A. O.
Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors
2015, vol. 20, no. 5, pp.  605-626
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 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, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 605-626
DOI:10.1134/S1560354715050056
Borisov A. V., Mamaev I. S., Bizyaev I. A.
The Jacobi Integral in Nonholonomic Mechanics
2015, vol. 20, no. 3, pp.  383-400
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.
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 Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 383-400
DOI:10.1134/S1560354715030107
Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Dynamics of Three Vortex Sources
2014, vol. 19, no. 6, pp.  694-701
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.
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, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 694-701
DOI:10.1134/S1560354714060070
Bizyaev I. A., Borisov A. V., Mamaev I. S.
Superintegrable Generalizations of the Kepler and Hook Problems
2014, vol. 19, no. 3, pp.  415-434
Abstract
In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Hook problems, both in two-dimensional spaces — the plane $\mathbb{R}^2$ and the sphere $S^2$ — and in three-dimensional spaces $\mathbb{R}^3$ and $S^3$. Using the central projection and the reduction procedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form.
Keywords: superintegrable systems, Kepler and Hook problems, isomorphism, central projection, reduction, highest degree polynomial superintegrals
Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  Superintegrable Generalizations of the Kepler and Hook Problems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 415-434
DOI:10.1134/S1560354714030095
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
2014, vol. 19, no. 2, pp.  198-213
Abstract
In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to 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 a nonholonomic hinge. 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. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.
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, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 198-213
DOI:10.1134/S156035471402004X
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
2013, vol. 18, no. 3, pp.  277-328
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 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, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 277-328
DOI:10.1134/S1560354713030064

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