Volume 22, Number 7
Volume 22, Number 7, 2017
Anatoly Pavlovich Markeev (on the occasion of his 75th birthday)
Markeev A. P.
On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourthorder Resonance
Abstract
The problem of orbital stability of a periodic motion of an autonomous twodegreeoffreedom Hamiltonian system is studied. The linearized equations of perturbed motion always have two real multipliers equal to one, because of the autonomy and the Hamiltonian structure
of the system. The other two multipliers are assumed to be complex conjugate numbers with absolute values equal to one, and the system has no resonances up to third order inclusive, but has a fourthorder resonance. It is believed that this case is the critical one for the resonance,
when the solution of the stability problem requires considering terms higher than the fourth degree in the series expansion of the Hamiltonian of the perturbed motion.
Using Lyapunov’s methods and KAM theory, sufficient conditions for stability and instability are obtained, which are represented in the form of inequalities depending on the coefficients of series expansion of the Hamiltonian up to the sixth degree inclusive. 
Cabral H. E., Amorim T. A.
Subharmonic Solutions of a Pendulum Under Vertical Anharmonic Oscillations of the Point of Suspension
Abstract
We prove the existence of subharmonic solutions in the dynamics of a pendulum whose point of suspension executes a vertical anharmonic oscillation of small amplitude.

Kholostova O. V., Safonov A. I.
A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance
Abstract
We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium
of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.

Bardin B. S., Chekina E. A.
On the Constructive Algorithm for Stability Analysis of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Secondorder Resonance Case
Abstract
This paper is concerned with a nonautonomous Hamiltonian system with two degrees of freedom whose Hamiltonian is a $2\pi$periodic function of time and analytic in a neighborhood of an equilibrium point. It is assumed that the system exhibits a secondorder resonance, i. e., the system linearized in a neighborhood of the equilibrium point has a double multiplier equal to $−1$. The case of general position is considered when the monodromy matrix is not reduced to diagonal form and the equilibrium point is linearly unstable. In this case, a nonlinear analysis is required to draw conclusions on the stability (or instability) of the equilibrium point in the complete system.
In this paper, a constructive algorithm for a rigorous stability analysis of the equilibrium point of the abovementioned system is presented. This algorithm has been developed on the basis of a method proposed in [1]. The main idea of this method is to construct and normalize a symplectic map generated by the phase flow of a Hamiltonian system. It is shown that the normal form of the Hamiltonian function and the generating function of the corresponding symplectic map contain no thirddegree terms. Explicit formulae are obtained which allow one to calculate the coefficients of the normal form of the Hamiltonian in terms of the coefficients of the generating function of a symplectic map. The developed algorithm is applied to solve the problem of stability of resonant rotations of a symmetric satellite. 
Iñarrea M., Lanchares V., Pascual A. I., Elipe A.
On the Stability of a Class of Permanent Rotations of a Heavy Asymmetric Gyrostat
Abstract
We consider the motion of an asymmetric gyrostat under the attraction of a uniform Newtonian field. It is supposed that the center of mass lies along one of the principal axes of inertia, while a rotor spins around a different axis of inertia. For this problem, we obtain the possible permanent rotations, that is, the equilibria of the system. The Lyapunov stability of these permanent rotations is analyzed by means of the Energy–Casimir method and necessary and sufficient conditions are derived, proving that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space. The geometry of the gyrostat and the value of the gyrostatic momentum are relevant in order to get stable permanent rotations. Moreover, it seems that the necessary conditions are also sufficient, but this fact can only be proved partially.

Maciejewski A. J., Przybylska M.
Global Properties of Kovalevskaya Exponents
Abstract
This paper contains a collection of properties of Kovalevskaya exponents which are eigenvalues of a linearization matrix of weighted homogeneous nonlinear systems along certain straightline particular solutions. Relations in the form of linear combinations of Kovalevskaya
exponents with nonnegative integers related to the presence of first integrals of the weighted homogeneous nonlinear systems have been known for a long time. As a new result other nonlinear relations between Kovalevskaya exponents calculated on all straightline particular
solutions are presented. They were obtained by an application of the Euler–Jacobi–Kronecker formula specified to an appropriate $n$form in a certain weighted homogeneous projective space

Churkina T. E., Stepanov S. Y.
On the Stability of Periodic Mercurytype Rotations
Abstract
We consider the stability of planar periodic Mercurytype rotations of a rigid body around its center of mass in an elliptical orbit in a central Newtonian field of forces. Mercurytype rotations mean that the body makes 3 turns around its center of mass during 2 revolutions of the center of mass in its orbit (resonance 3:2). These rotations can be 1) symmetrical $2\pi$periodic, 2) symmetrical $4\pi$periodic and 3) asymmetrical $4\pi$periodic. The stability of rotations of type 1) was investigated by A.P. Markeev. In our paper we present a nonlinear stability analysis for some rotations of types 2) and 3) in 3rd and 4thorder resonant cases, in the nonresonant case and at the boundaries of regions of linear stability.

Kurakin L. G., Ostrovskaya I. V.
On Stability of Thomson’s Vortex $N$gon in the Geostrophic Model of the Point Bessel Vortices
Abstract
A stability analysis of the stationary rotation of a system of $N$ identical point Bessel vortices lying uniformly on a circle of radius $R$ is presented. The vortices have identical intensity $\Gamma$ and length scale $\gamma^{1}>0$. The stability of the stationary motion is interpreted as equilibrium stability of a reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied.
The cases for $N=2,\ldots,6$ are studied sequentially. The case of odd $N=2\ell+1\geqslant 7$ vortices and the case of even $N=2n\geqslant 8$ vortices are considered separately. It is shown that the $(2\ell+1)$gon is exponentially unstable for $0 < \gamma R < R_*(N)$. However, this $(2\ell+1)$gon is stable for $\gamma R\geqslant R_*(N)$ in the case of the linearized problem (the eigenvalues of the linearization matrix lie on the imaginary axis). The even $N=2n\geqslant 8$ vortex $2n$gon is exponentially unstable for $R>0$.

Gutierres R., Vidal C.
Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case
Abstract
This paper concerns with the study of the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a
neighborhood of the equilibrium point with $1$degree of freedom in the degenerate case $H= q^4+ H_5+ H_6+\cdots$. Our main results complete the study initiated by Markeev in [9].
