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    Alexandr Burov

    117967, Moscow, Vavilova str.,40
    Dorodnicyn Computing Cenyer of the RAS


    Burov A. A., Guerman A., Raspopova E., Nikonov V.
    It is well known that several small celestial objects are of irregular shape. In particular, there exist asteroids of the so-called “dog-bone” shape. It turns out that approximation of these bodies by dumb-bells, as proposed by V.V. Beletsky, provides an effective tool for analytical investigation of dynamics in vicinities of such bodies. There remains the question of how to divide reasonably a “dogbone” body into two parts using available measurement data.
    In this paper we introduce an approach based on the so-called $K$-mean algorithm proposed by the prominent Polish mathematician H. Steinhaus.
    Keywords: $K$-means algorithm, small celestial bodies, mesh representation of an asteroid’s surface
    Citation: Burov A. A., Guerman A., Raspopova E., Nikonov V.,  On the use of the $K$-means algorithm for determination of mass distributions in dumbbell-like celestial bodies, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp.  45-52
    Burov A. A., Kosenko I.
    Within the framework of the so-called satellite approximation, configurations of the relative equilibrium are built and their stability is analyzed. In this case the elliptic Keplerian motion of the satellite/the spacecraft tight group mass center is predefined. The attitude motion of the system does not influence its orbital motion. The principal central axes of inertia are assumed to move as a rigid body. Simultaneously masses of the body can redistribute in a way such that the values of moments of inertia can change. Thus, all configurations can perform pulsing motions changing it own dimensions.
    One obtains a system of equations of motion for such a compound satellite. It turns out that the resulting system of equations is similar to the well-known equation of V.V.Beletsky for the satellite in elliptic orbit planar oscillations. We use true anomaly as an independent variable as it is in the Beletsky equation. It turned out that there are planar pendulum-like librations of the whole system which may be regarded as perturbations of the mathematical pendulum.
    One can introduce action-angle variables in this case and can construct the dynamics of mappings over the non-autonomous perturbation period. As a result, one is able to apply the well-known Moser theorem on an invariant curve for twisting maps of annulus. After that one can get a general picture of motion in the case of the system planar oscillations. So, the whole description in the paper splits into two topics: (a) general dynamical analysis of the satellite planar attitude motion using KAM theory; (b) construction of periodic solutions families depending on the perturbation parameter and rising from equilibrium as the perturbation value grows. The latter families depend on the parameter of the perturbation and are absent in the non-perturbed problem.
    Keywords: KAM theory, Moser theorem on invariant curve, action-angle variables, periodic solutions, analytical developments
    Citation: Burov A. A., Kosenko I.,  Motion of a satellite with a variable mass distribution in a central field of Newtonian attraction, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 4, pp.  519–531
    Burov A. A., Guerman A., Kosenko I., Nikonov V.
    The problem of the motion of a particle in the gravity field of a homogeneous dumbbell-like body composed of a pair of intersecting balls, whose radii are, in general, different, is studied. Approximation for the Newtonian potential of attraction is obtained. Relative equilibria and their properties are studied under the assumption of uniform rotation of the dumbbells.
    Keywords: generalized planar two-bodies problem, asteroid-like systems, gravitating systems with irregular mass distribution, stability of steady motions, bifurcations of steady motions
    Citation: Burov A. A., Guerman A., Kosenko I., Nikonov V.,  On the gravity of dumbbell-like bodies represented by a pair of intersecting balls, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 2, pp.  243-256
    Burov A. A., Nikonov V.
    The planar motion of an equilateral triangle with equal masses at vertices and of a point subjected to mutual Newtonian attraction is considered. Necessary conditions for the stability of “straight”, axial steady configurations, when the massive point is located on one of the symmetry axes of the triangle, are studied. The generation of other, “oblique”, steady configurations is discussed in connection with the variation, for certain parameter values, of the degree of instability of some “straight” steady configurations.
    Keywords: generalized planar two-body problem, asteroid-like systems, gravitating systems with irregular mass distribution, stability of steady motions, necessary conditions for stability, gyroscopic stabilization, bifurcations of steady motions, Poincaré bifurcation diagrams
    Citation: Burov A. A., Nikonov V.,  Stability and branching of stationary rotations in a planar problem of motion of mutually gravitating triangle and material point, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 2, pp.  179-196

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