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2013
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# Ivan Bizyaev

ul. Universitetskaya 1, Izhevsk, 426034 Russia
Udmurt State University

Research assistant of the Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles at UdSU

## Publications:

 Bizyaev I. A., Borisov A. V., Mamaev I. S. Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem 2019, vol. 24, no. 5, pp.  560-582 Abstract This paper addresses the problem of the rolling of a spherical shell with a frame rotating inside, on which rotors are fastened. It is assumed that the center of mass of the entire system is at the geometric center of the shell. For the rubber rolling model and the classical rolling model it is shown that, if the angular velocities of rotation of the frame and the rotors are constant, then there exists a noninertial coordinate system (attached to the frame) in which the equations of motion do not depend explicitly on time. The resulting equations of motion preserve an analog of the angular momentum vector and are similar in form to the equations for the Chaplygin ball. Thus, the problem reduces to investigating a two-dimensional Poincaré map. The case of the rubber rolling model is analyzed in detail. Numerical investigation of its Poincaré map shows the existence of chaotic trajectories, including those associated with a strange attractor. In addition, an analysis is made of the case of motion from rest, in which the problem reduces to investigating the vector field on the sphere $S^2$. Keywords: nonholonomic mechanics, Chaplygin ball, rolling without slipping and spinning, strange attractor, straight-line motion, stability, limit cycle, balanced beaver-ball Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem, Regular and Chaotic Dynamics, 2019, vol. 24, no. 5, pp. 560-582 DOI:10.1134/S1560354719050071
 Bizyaev I. A., Borisov A. V., Mamaev I. S. Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control 2018, vol. 23, no. 7-8, pp.  983-994 Abstract In this paper we consider the problem of the motion of the Roller Racer.We assume that the angle $\varphi (t)$ between the platforms is a prescribed function of time. We prove that in this case the acceleration of the Roller Racer is unbounded. In this case, as the Roller Racer accelerates, the increase in the constraint reaction forces is also unbounded. Physically this means that, from a certain instant onward, the conditions of the rolling motion of the wheels without slipping are violated. Thus, we consider a model in which, in addition to the nonholonomic constraints, viscous friction force acts at the points of contact of the wheels. For this case we prove that there is no constant acceleration and all trajectories of the reduced system asymptotically tend to a periodic solution. Keywords: Roller Racer, speed-up, nonholonomic mechanics, Rayleigh dissipation function, viscous friction, integrability by quadratures, control, constraint reaction force Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 983-994 DOI:10.1134/S1560354718070122
 Bizyaev I. A., Borisov A. V., Mamaev I. S. An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane 2018, vol. 23, no. 6, pp.  665-684 Abstract This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero. Keywords: nonholonomic mechanics, regularization, blowing-up, invariant measure, ergodic theorems, normal hyperbolic submanifold, Poincaré map, first integrals Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane, Regular and Chaotic Dynamics, 2018, vol. 23, no. 6, pp. 665-684 DOI:10.1134/S1560354718060035
 Borisov A. V., Mamaev I. S., Bizyaev I. A. Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability 2018, vol. 23, no. 5, pp.  613-636 Abstract This paper is concerned with the problem of three vortices on a sphere $S^2$ and the Lobachevsky plane $L^2$. After reduction, the problem reduces in both cases to investigating a Hamiltonian system with a degenerate quadratic Poisson bracket, which makes it possible to study it using the methods of Poisson geometry. This paper presents a topological classification of types of symplectic leaves depending on the values of Casimir functions and system parameters. Keywords: Poisson geometry, point vortices, reduction, quadratic Poisson bracket, spaces of constant curvature, symplectic leaf, collinear configurations Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A.,  Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability, Regular and Chaotic Dynamics, 2018, vol. 23, no. 5, pp. 613-636 DOI:10.1134/S1560354718050106
 Bizyaev I. A., Borisov A. V., Mamaev I. S. The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration 2017, vol. 22, no. 8, pp.  955–975 Abstract This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found. Keywords: nonholonomic mechanics, Fermi acceleration, Chaplygin sleigh, parametric oscillator, tensor invariants, involution, strange attractor, Lyapunov exponents, reversible systems, chaotic dynamics Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 955–975 DOI:10.1134/S1560354717080056
 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