Olga Kholostova
Volokolamskoe Shosse 4, 125993,GSP3, Moscow, Russia
Moscow Aviation Institute (National Research University)
Publications:
Kholostova O. V.
Abstract
This paper is concerned with the motions of a nearautonomous twodegreeoffreedom
Hamiltonian system, $2\pi$periodic in time, in a neighborhood of a trivial equilibrium. It is assumed
that in the autonomous case, in the region where only necessary (which are not sufficient)
conditions for the stability of this equilibrium are satisfied, for some parameter values of the
system one of the frequencies of small linear oscillations is equal to two and the other is equal
to one. An analysis is made of nonlinear oscillations of the system in a neighborhood of this
equilibrium for the parameter values near a resonant point of parameter space. The boundaries
of the parametric resonance regions are constructed which arise in the presence of secondary
resonances in the transformed linear system (the cases of zero frequency and equal frequencies).
The general case and both cases of secondary resonances are considered; in particular, the case
of two zero frequencies is singled out. An analysis is made of resonant periodic motions of the
system that are analytic in integer or fractional powers of the small parameter, and conditions
for their linear stability are obtained. Using KAM theory, two and threefrequency conditionally
periodic motions (with frequencies of different orders in a small parameter) are described.

Kholostova O. V.
Abstract
We consider the motions of a nearautonomous Hamiltonian system $2\pi$periodic in time,
with two degrees of freedom, in a neighborhood of a trivial equilibrium. A multiple parametric
resonance is assumed to occur for a certain set of system parameters in the autonomous case,
for which the frequencies of small linear oscillations are equal to two and one, and the resonant
point of the parameter space belongs to the region of sufficient stability conditions. Under certain
restrictions on the structure of the Hamiltonian of perturbed motion, nonlinear oscillations of
the system in the vicinity of the equilibrium are studied for parameter values from a small
neighborhood of the resonant point. Analytical boundaries of parametric resonance regions are
obtained, which arise in the presence of secondary resonances in the transformed linear system
(the cases of zero frequency and equal frequencies). The general case, for which the parameter
values do not belong to the parametric resonance regions and their small neighborhoods, and
both cases of secondary resonances are considered. The question of the existence of resonant
periodic motions of the system is solved, and their linear stability is studied. Two and threefrequency
conditionally periodic motions are described. As an application, nonlinear resonant
oscillations of a dynamically symmetric satellite (rigid body) relative to the center of mass in
the vicinity of its cylindrical precession in a weakly elliptical orbit are investigated.

Kholostova O. V.
Abstract
This paper examines the motion of a timeperiodic Hamiltonian system with two degrees
of freedom in a neighborhood of trivial equilibrium. It is assumed that the system depends
on three parameters, one of which is small; when it has zero value, the system is autonomous.
Consideration is given to a set of values of the other two parameters for which, in the autonomous
case, two frequencies of small oscillations of the linearized equations of perturbed motion are
identical and are integer or halfinteger numbers (the case of multiple parametric resonance).
It is assumed that the normal form of the quadratic part of the Hamiltonian does not reduce to
the sum of squares, i.e., the trivial equilibrium of the system is linearly unstable. Using a number
of canonical transformations, the perturbed Hamiltonian of the system is reduced to normal form
in terms through degree four in perturbations and up to various degrees in a small parameter
(systems of first, second and third approximations). The structure of the regions of stability and
instability of trivial equilibrium is investigated, and solutions are obtained to the problems of
the existence, number, as well as (linear and nonlinear) stability of the system’s periodic motions
analytic in fractional or integer powers of the small parameter. For some cases, conditionally
periodic motions of the system are described. As an application, resonant periodic motions of
a dynamically symmetric satellite modeled by a rigid body are constructed in a neighborhood
of its steady rotation (cylindrical precession) on a weakly elliptic orbit and the problem of their
stability is solved.

Kholostova O. V.
Abstract
The motion of a solid (satellite) carrying a moving point mass in the central Newtonian
gravitational field in an elliptical orbit of arbitrary eccentricity is considered. The law of motion
of a point mass is assumed to allow for the existence of relative equilibria of the “bodypoint”
system in the orbital coordinate system. A nonlinear stability analysis of these equilibria is carried
out, based on the construction and normalization of the areapreserving mapping generated
by the motions of the system.

Kholostova O. V.
Abstract
The motion of a heavy rigid body with a mass geometry corresponding to the Hess case is
considered. The suspension point of the body is assumed to perform highfrequency periodic
vibrations of small amplitude in the threedimensional space. It is proved that for any law of
vibrations of this type, the approximate autonomous equations of the body motion admit an
invariant relation (the first integral at the zero level), which coincides with a similar relation that
exists in the Hess case of the motion of a body with a fixed point. In the approximate equations
of motion written in Hamiltonian form, the cyclic coordinate is introduced and the corresponding
reduction is performed. For the laws of vibration of the suspension point corresponding to the
integrable cases (when there is another cyclic coordinate in the system), a detailed study of
the model onedegreeoffreedom system is given. For the nonintegrable cases, an analogy with
the approximate problem of the motion of a Lagrange top with a vibrating suspension point is
drawn, and the results obtained earlier for the top are used. Some properties of the body motion
at the nonzero level of the above invariant relation are also discussed.

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.

Belichenko M. V., Kholostova O. V.
Abstract
We consider the motion of Lagrange’s top with a suspension point performing the specified highfrequency periodic motion with small amplitude in threedimensional 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 twodegreeoffreedom 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.

Kholostova O. V.
Abstract
The motion of a timeperiodic twodegreeoffreedom Hamiltonian system in the neighborhood of the equilibrium being stable in the linear approximation is considered. The weak Raman thirdorder resonance and the strong fourthorder resonance are assumed to occur simultaneously in the system. The behavior of the approximated (model) system is studied in the stability domain of the fourthorder 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.

Kholostova O. V.
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.

Kholostova O. V.
Abstract
Motions of a timeperiodic, twodegreeoffreedom 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 thirdorder 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.

Kholostova O. V.
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 fivedimensional 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.

Kholostova O. V.
Abstract
Motions of a nonautonomous timeperiodic 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 KAMtheory 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.

Kholostova O. V.
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 pendulumlike 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.
