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2013
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# Olga Kholostova

Volokolamskoe Shosse 4, 125993,GSP-3, Moscow, Russia
Moscow Aviation Institute

## Publications:

 Kholostova O. V. On periodic motions of a nonautonomous Hamiltonian system in one case of multiple parametric resonance 2017, Vol. 13, No. 4, pp.  477–504 Abstract The motion of a nonautonomous time-periodic two-degree-of-freedom 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 half-integer 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. Keywords: Hamiltonian system, multiple parametric resonance, periodic motions, stability, cylindrical precession of a satellite Citation: Kholostova O. V.,  On periodic motions of a nonautonomous Hamiltonian system in one case of multiple parametric resonance, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 4, pp.  477–504 DOI:10.20537/nd1704003
 Belichenko M. V., Kholostova O. V. On the stability of stationary rotations in the approximate problem of motion of Lagrange’s top with a vibrating suspension point 2017, Vol. 13, No. 1, pp.  81-104 Abstract We consider the motion of Lagrange’s top with a suspension point performing the specified highfrequency periodic motion with small amplitude in three-dimensional space. The approximate autonomous system of equations of motion written in the form of canonical Hamiltonian equations is investigated. The problem of the existence and number of stationary rotations of the top about its dynamical symmetry axis is solved. The study of stability of the corresponding equilibrium positions of the reduced two-degree-of-freedom system for fixed values of the cyclic integral constant depending on the angular velocity of rotation is carried out. For suspension points’ motions allowing for stationary rotations about the vertical, a detailed linear and nonlinear stability analysis of these rotations and rotations about inclined axes is carried out. For a number of other cases of the suspension point motions a linear stability analysis is carried out. Keywords: Lagrange’s top, “sleeping” top, high-frequency vibrations, stability Citation: Belichenko M. V., Kholostova O. V.,  On the stability of stationary rotations in the approximate problem of motion of Lagrange’s top with a vibrating suspension point, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 1, pp.  81-104 DOI:10.20537/nd1701006
 Kholostova O. V. The interaction of resonances of the third and fourth orders in a Hamiltonian two-degree-of-freedom system 2015, Vol. 11, No. 4, pp.  671–683 Abstract The motion of a time-periodic two-degree-of-freedom Hamiltonian system in the neighborhood of the equilibrium being stable in the linear approximation is considered. The weak Raman thirdorder resonance and the strong fourth-order resonance are assumed to occur simultaneously in the system. The behavior of the approximated (model) system is studied in the stability domain of the fourth-order resonance. Areas of the parameters (coefficients of the normalized Hamiltonian) are found for which all motions of the system are bounded if they begin in a sufficiently small neighborhood of the equilibrium. Boundedness domain estimate is obtained. А disturbing effect of the double resonance on the motion of the system within the boundedness domain is described. Keywords: Hamiltonian system, canonical transformation, method of normal forms, double resonance, stability Citation: Kholostova O. V.,  The interaction of resonances of the third and fourth orders in a Hamiltonian two-degree-of-freedom system, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 4, pp.  671–683 DOI:10.20537/nd1504004
 Kholostova O. V. On the stability of the specific motions of a heavy rigid body due to fast vertical vibrations of one of its points 2015, Vol. 11, No. 1, pp.  99-116 Abstract We consider the motion of a heavy rigid body with one point performing the specified highfrequency harmonic oscillations along the vertical. In the framework of an approximate autonomous system of differential equations of motion two new types of permanent rotations of the body about the vertical are found. These motions are affected by presence of fast vibrations and do not exist in the case of a body with a fixed point. The problem of stability of the motions is investigated. Keywords: rigid body, fast vibrations, permanent rotations, stability, resonance Citation: Kholostova O. V.,  On the stability of the specific motions of a heavy rigid body due to fast vertical vibrations of one of its points, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 1, pp.  99-116 DOI:10.20537/nd1501005
 Kholostova O. V. Motions of a two-degree-of-freedom Hamiltonian system in the presence of multiple third-order resonances 2012, Vol. 8, No. 2, pp.  267-288 Abstract Motions of a time-periodic, two-degree-of-freedom Hamiltonian system in a neighborhood of a linearly stable equilibrium are considered. It is assumed that there are several resonant thirdorder relations between the frequencies of linear oscillations of the system. It is shown that in the presence of two third-order resonances the equilibrium is unstable at any ratio between resonant coefficients. Approximate (model) Hamiltonians are obtained which are characteristic of the resonant cases under consideration. A detailed analysis is made of nonlinear oscillations of systems corresponding to them. Keywords: Hamiltonian system, multiple resonance, stability, Chetaev function Citation: Kholostova O. V.,  Motions of a two-degree-of-freedom Hamiltonian system in the presence of multiple third-order resonances, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 2, pp.  267-288 DOI:10.20537/nd1202005
 Kholostova O. V. On stability of permanent Staude's rotations in a general case of a mass geometry of a rigid body 2009, Vol. 5, No. 3, pp.  357-375 Abstract Stability of permanent rotations around the vertical of a heavy rigid body with the immovable point (Staude’s rotations) is investigated in assumption of a general mass distribution in the body and an arbitrary position of the point of support. In admissible domains of the five-dimensional space of parameters of the problem the detailed linear analysis of stability is carried out. For each set of admissible values of parameters the necessary conditions of stability are received. In a number of cases the sufficient conditions of stability are found. Keywords: Euler-Poisson's equations, permanent rotations, cone of Staude, stability Citation: Kholostova O. V.,  On stability of permanent Staude's rotations in a general case of a mass geometry of a rigid body, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 3, pp.  357-375 DOI:10.20537/nd0903004
 Kholostova O. V. Abstract Motions of a non-autonomous time-periodic Hamiltonian system with one degree of freedom are considered. The Hamiltonian of the system contains a small parameter. The origin of the phase space is a linearly stable equilibrium of the unperturbed or complete system. It is supposed that the degeneration takes place in the unperturbed system with regard for terms of order less than five (the frequency of small nonlinear oscillations does not depend on the amplitude), and a resonance (up to the sixth order inclusively) occurs. For each resonance case a model Hamiltonian is constructed, and a qualitative investigation of motion of the model system is carried out. Using Poincare’s theory of periodic motions and KAM-theory we solve rigorously the problem of existence, bifurcations and stability of periodic motions of the initial system. The motions we study are analytical with respect to fractional (for resonances up to the forth order inclusively) or integer (resonances of fifth and sixth orders) degrees of the small parameter. As an illustration, we analyze resonance periodic motions of a spherical pendulum and a Lagrange top with a vibrating point of suspension in the presence of the degeneration considered. Keywords: hamiltonian system, stability, resonance, Poincare's theory of periodical motions, KAM-theory Citation: Kholostova O. V.,  On bifurcations and stability of resonance periodic motions of hamiltonian systems with one degree of freedom caused by degeneration of the hamiltonian, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 1, pp.  89-110 DOI:10.20537/nd0601005
 Kholostova O. V. Linear analysis of stability the planar oscillations of a satellite being a plate in a circular orbit 2005, Vol. 1, No. 2, pp.  181-190 Abstract We study the motion of a satellite (a rigid body) in a circular orbit about its centre of mass. The satellite is subject to the central Newtonian gravitational field. The satellite’s principal central moments of inertia $A$, $B$ and $C$ are assumed to satisfy the equation $B=A+C$. This equation holds for thin plates. Particular motions occur when the plate executes pendulum-like oscillations of an arbitrary amplitude in the plane of the orbit. A linear analysis of the orbital stability of this motion is carried out. In the plane of parameters of the problem (an amplitude of oscillations and an inertial parameter) domains of orbital linear stability and instability of oscillations of the satellite are obtained both numerically and analytically. Keywords: satellite, orbital stability, parametric resonance, action-angle variables, Deprit-Hori method Citation: Kholostova O. V.,  Linear analysis of stability the planar oscillations of a satellite being a plate in a circular orbit, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp.  181-190 DOI:10.20537/nd0502002