Vol. 13, No. 4
Vol. 13, No. 4, 2017
On the 75th birthday of A.P.Markeev
Bardin B. S., Chekina E. A.
On the stability of planar oscillations of a satelliteplate in the case of essential type resonance
Abstract
We consider satellite motion about its center of mass in a circle orbit. We study the problem of orbital stability for planar pendulumlike oscillations of the satellite. It is supposed that the satellite is a rigid body whose mass geometry is that of a plate. We assume that on the unperturbed motion the middle or minor inertia axis of the satellite lies in the orbit plane, i.e., the plane of the satelliteplate is perpendicular to the plane of the orbit. In this paper we perform a nonlinear analysis of the orbital stability of planar pendulumlike oscillations of a satelliteplate for previously unexplored parameter values corresponding to the boundaries of regions of stability in the first approximation, where the essential type resonances take place. It is proved that on the mentioned boundaries the planar pendulumlike oscillations are formally orbital stable or orbitally stable in third approximation. 
Kholostova O. V.
Abstract
The motion of a nonautonomous timeperiodic twodegreeoffreedom Hamiltonian system in a neighborhood of an equilibrium point is considered. The Hamiltonian function of the system is supposed to depend on two parameters $\varepsilon$ and $\alpha$, with $\varepsilon$ being small and the system being autonomous at $\varepsilon=0$. It is also supposed that for $\varepsilon=0$ and some values of $\alpha$ one of the frequencies of small linear oscillations of the system in the neighborhood of the equilibrium point is an integer or halfinteger and the other is equal to zero, that is, the system exhibits a multiple parametric resonance. The case is considered where the rank of the matrix of equations of perturbed motion that are linearized at $\varepsilon=0$ in the neighborhood of the equilibrium point is equal to three. For sufficiently small but nonzero values of $\varepsilon$ and for values of $\alpha$ close to the resonant ones, the question of existence, bifurcations, and stability (in the linear approximation) of the periodic motions of the system is solved. As an application, periodic motions of a symmetrical satellite in the neighborhood of its cylindrical precession in an orbit with small eccentricity are constructed for cases of the multiple resonances considered.

Rodnikov A. V., Krasil'nikov P. S.
Abstract
We study motions of a particle along a rope with ends fixed to an extended rigid body whose center of mass traces out a circular orbit in the central Newtonian force field. (Such a rope is called a tether.) We assume that the tether realizes an ideal unilateral constraint. We derive particle motion equations on the surface of the ellipsoid, which restricts the particle motion, and conditions that guarantee such motions. We also study the existence and stability of relative equilibria of the particle with respect to the orbital frame of reference. We prove stability of the integral manifold of the particle motions in the plane of the orbit. We note that smallamplitude librations near this manifold can be described by approximate equations that can be reduced to Riccati’s equation. We establish that generally the spacial motions of the particle are chaotic for initial conditions from some vicinity of the separatrix motion in the plane of the orbit and are regular in other cases. We also note that chaotic motions usually lead to a situation where the particle comes off the constraint, in other words, to motions inside the abovementioned ellipsoid.

Burov A. A., Kosenko I.
Abstract
Within the framework of the socalled 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 wellknown 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 pendulumlike librations of the whole system which may be regarded as perturbations of the mathematical pendulum. One can introduce actionangle variables in this case and can construct the dynamics of mappings over the nonautonomous perturbation period. As a result, one is able to apply the wellknown 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 nonperturbed problem. 
Baikov A., Kovalev N.
Abstract
The motion of a piecewise linear oscillator is considered. It consists of two spring connected drawers on a conveyor belt moving at a constant speed. The equations of motion are averaged in one nonresonance case. A continuum of invariant tori is obtained that exists in the exact system. The attraction (in finite time) of the trajectories to the family of limit tori is proved (limit tori belong to the continuum of invariant tori). We also investigate zones of sticking, which cannot be detected by averaging.

Krasil'nikov P. S.
Abstract
The plane circular restricted threebody problem is considered, where the massless body is a constant lowthrust spacecraft. It is assumed that the vector of lowthrust is directed along the $Ox$ axis connecting the main bodies. The problem of plotting a family of oneparameter Hill’s curves is investigated. The existence conditions of artificial triangulartype and collineartype libration points are obtained. The values of the effective force function at libration points are investigated also. Six different topological types of the family of oneparameter Hill’s curves are described. It is shown that these types differ in the number of critical values of the constant Jacobi integral and in the ordering of these values. For the Earth – Moon system, a family of oneparameter Hill’s curves is plotted for each of the six types.

Grines V. Z., Kurenkov E. D.
Abstract
It is well known that the topological classification of dynamical systems with hyperbolic dynamics is significantly defined by dynamics on a nonwandering set. F. Przytycki generalized axiom $A$ for smooth endomorphisms that was previously introduced by S. Smale for diffeomorphisms, and proved the spectral decomposition theorem which claims that the nonwandering set of an $A$endomorphism is a union of a finite number of basic sets. In the present paper the criterion for a basic set of an $A$endomorphism to be an attractor is given. Moreover, dynamics on basic sets of codimension one is studied. It is shown that if an attractor is a topological submanifold of codimension one of type $(n − 1, 1)$, then it is smoothly embedded in the ambient manifold, and the restriction of the endomorphism to this basic set is an expanding endomorphism. If a basic set of type $(n, 0)$ is a topological submanifold of codimension one, then it is a repeller, and the restriction of the endomorphism to this basic set is also an expanding endomorphism. 
Pochinka O. V., Kruglov E. V., Dolgonosova A.
Abstract
In this paper, one of the possible scenarios for the creation of heteroclinic separators in the solar corona is described and realized. This reconnection scenario connects the magnetic field with two zero points of different signs, the fan surfaces of which do not intersect, with a magnetic field with two zero points which are connected by two heteroclinic separators. The method of proof is to create a model of the magnetic field produced by the plasma in the solar corona and to study it using the methods of dynamical systems theory. Namely, in the space of vector fields on the sphere $S^3$ with two sources, two sinks and two saddles, we construct a simple arc with two saddlenode bifurcation points that connects the system without heteroclinic curves to a system with two heteroclinic curves. The discretization of this arc is also a simple arc in the space of diffeomorphisms. The results are new.

Zhukova N. I.
Abstract
In this paper we present an explicit construction of a continuum family of smooth pairwise nonisomorphic foliations of codimension one on a standard threedimensional sphere, each of which has a countable set of compact attractors which are leaves diffeomorphic to a torus. As it was proved by S.P.Novikov, every smooth foliation of codimension one on a standard threedimensional sphere contains a Reeb component. Changing this foliation only in the Reeb component by the method presented, we get a continuum family of smooth pairwise nonisomorphic foliations containing a countable set of compact attractor leaves diffeomorphic to a torus which coincides with the original foliation outside this Reeb component.

Vetchanin E. V., Klenov A. I.
Abstract
This paper presents a comparative analysis of computations of the motion of heavy threebladed screws in a fluid along with experimental results. Simulation of the motion is performed using the theory of an ideal fluid and the phenomenological model of viscous friction. For experimental purposes, models of threebladed screws with various configurations and sizes were manufactured by casting from chemically hardening polyurethane. Comparison of calculated and experimental results has shown that the mathematical models considered essentially do not reflect the processes observed in the experiments.

Karavaev Y. L., Klekovkin A. V., Kilin A. A.
Abstract
In this paper the model of rolling of spherical bodies on a plane without slipping is presented taking into account viscous rolling friction. Results of experiments aimed at investigating the influence of friction on the dynamics of rolling motion are presented. The proposed dynamical friction model for spherical bodies is verified and the limits of its applicability are estimated. A method for determining friction coefficients from experimental data is formulated.

Pivovarova E. N.
Abstract
The dynamics of a spherical robot of combined type consisting of a spherical shell and a pendulum attached at the center of the shell is considered. At the end of the pendulum a rotor is installed. For this system we carry out a stability analysis for a partial solution which in absolute space corresponds to motion along a circle with constant velocity. Regions of stability of a partial solution are found depending on the orientation of the spherical robot during the motion, its velocity and the radius of the circle traced out by the point of contact.

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.
