Alexey Bolsinov

    Alexey Bolsinov
    Ashby Road, Loughborough, LE11 3TU, UK
    Loughborough University

    D.Sc., Professor

    Member of the editorial board of «Regular & Chaotic Dynamics»


    Bizyaev I. A.,  Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S.
    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
    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
    Bolsinov A. V.,  Kilin A. A.,  Kazakov A. O.
    The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.
    Keywords: topological monodromy, integrable systems, nonholonomic systems, Poincaré map, bifurcation analysis, focus-focus singularities
    Citation: Bolsinov A. V.,  Kilin A. A.,  Kazakov A. O., Topological monodromy in nonholonomic systems, Rus. J. Nonlin. Dyn., 2013, Vol. 9, no. 2, pp. 203-227
    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
    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
    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

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