Boris Bardin

A general method is presented for constructing a nonlinear canonical transformation, which makes it possible to introduce local variables in a neighborhood of periodic motions of an autonomous Hamiltonian system with two degrees of freedom. This method can be used for investigating the behavior of the Hamiltonian system in the vicinity of its periodic trajectories. In particular, it can be applied to solve the problem of orbital stability of periodic motions.

We consider the stability of the equilibrium position of a periodic Hamiltonian system with one degree of freedom. It is supposed that the series expansion of the Hamiltonian function, in a small neighborhood of the equilibrium position, does not include terms of second and third degree. Moreover, we focus on a degenerate case, when fourth-degree terms in the Hamiltonian function are not enough to obtain rigorous conclusions on stability or instability. A complete study of the equilibrium stability in the above degenerate case is performed, giving sufficient conditions for instability and stability in the sense of Lyapunov. The above conditions are expressed in the form of inequalities with respect to the coefficients of the Hamiltonian function, normalized up to sixth-degree terms inclusive.

We deal with a 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 characteristic equation of the system linearized in a neighborhood of the equilibrium point has two different double roots such that their absolute values are equal to unity, i.\,e., a combinational resonance takes place in this system. We consider the case of general position when the monodromy matrix of the linearized system is not diagonalizable. In this case the equilibrium point is linearly unstable. However, this does not imply its instability in the original nonlinear system. Rigorous conclusions on the stability can be formulated in terms of coefficients of the Hamiltonian normal form.
We describe a constructive algorithm for constructing and normalizing the symplectic map generated by the phase flow of the Hamiltonian system considered. We obtain explicit relations between the coefficients of the generating function of the symplectic map and the coefficients of the Hamiltonian normal form. It allows us to formulate conditions of stability and instability in terms of coefficients of the above generating function. The developed algorithm is applied to solve the stability problem for oscillations of a satellite with plate mass geometry, that is, $J_z = J_x +J_y$, where $J_x$, $J_y$, $J_z$ are the principal moments of inertia of the satellite, when the parameter values belong to a boundary of linear stability.

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 above-mentioned 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 third-degree 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.

We deal with the stability problem of resonant rotation of a symmetric rigid body about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the body completes one rotation in absolute space during two orbital revolutions of its center of mass. In [1–3] the stability analysis of the above resonant rotation with respect to planar perturbations has been performed in detail.
In this paper we study the stability of the resonant rotation in an extended formulation taking into account both planar and spatial perturbations. By analyzing linearized equations of perturbed motion, we found eccentricity intervals, where the resonant rotation is unstable. Outside of these intervals a nonlinear stability study has been performed and subintervals of formal stability and stability for most initial data have been found. In addition, the instability of the resonant rotation was established at several eccentricity values corresponding to the third and fourth order resonances.
Our study has also shown that in linear approximation the spatial perturbations have no effect on the stability of the resonant rotation, whereas in a nonlinear system they can lead to its instability at some resonant values of the eccentricity.

We deal with the stability problem of an equilibrium position of a periodic Hamiltonian system with one degree of freedom. We suppose the Hamiltonian is analytic in a small neighborhood of the equilibrium position, and the characteristic exponents of the linearized system have zero real part, i.e., a nonlinear analysis is necessary to study the stability in the sense of Lyapunov. In general, the stability character of the equilibrium depends on nonzero terms of the lowest order $N (N >2)$ in the Hamiltonian normal form, and the stability problem can be solved by using known criteria.
We study the so-called degenerate cases, when terms of order higher than $N$ must be taken into account to solve the stability problem. For such degenerate cases, we establish general conditions for stability and instability. Besides, we apply these results to obtain new stability criteria for the cases of degeneracy, which appear in the presence of first, second, third and fourth order resonances.

We study the Lyapunov stability problem of the resonant rotation of a rigid body satellite about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the satellite completes one rotation in absolute space during two orbital revolutions of its center of mass. The stability analysis of the above resonance rotation was started in [4, 6]. In the present paper, rigorous stability conclusions in the previously unstudied range of parameter values are obtained. In particular, new intervals of stability are found for eccentricity values close to 1. In addition, some special cases are studied where the stability analysis should take into account terms of degree not less than six in the expansion of the Hamiltonian of the perturbed motion. Using the technique described in [7, 8], explicit formulae are obtained, allowing one to verify the stability criterion of a time-periodic Hamiltonian system with one degree of freedom in the special cases mentioned.

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev–Steklov case. The stability problem is solved in nonlinear setting.
In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically.
In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients.

We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position.
Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of the perturbed motion are obtained in Hamiltonian form. Regions of orbital instability are established by means of linear analysis. Outside the above-mentioned regions, nonlinear analysis is performed taking into account terms up to degree 4 in the expansion of the Hamiltonian in a neighborhood of unperturbed motion. The nonlinear problem of orbital stability is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. The orbital stability is investigated analytically in two limiting cases: small amplitude oscillations and rotations with large angular velocities when a small parameter can be introduced.

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that the geometry of the mass of the body corresponds to the Bobylev–Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the basis of a nonlinear analysis.
In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities we study the problem analytically. In the general case we reduce the problem to the stability study of a fixed point of the symplectic map generated by equations of perturbed motion. We calculate coefficients of the symplectic map numerically. By analyzing the abovementioned coefficients we establish the orbital stability or instability of the unperturbed motion. The results of the study are represented in the form of a stability diagram.

We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin, which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3:1. We study nonlinear conditionally periodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the so-called truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionally periodic. By using the KAM theory methods we show that most of the conditionally periodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that are not conditionally periodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession.

We consider a point moving in an ellipsoid $a_1x_1^2+a_2x_2^2+a_3x_3^2=1$ under the influence of a force with quadratic potential $V=\frac{1}{2}(b_1x_1^2+b_2x_2^2+b_3x_3^2)$. We prove that the equations of motion of the point are meromorphically integrable if and only if the condition $b_1(a_2-a_3)+b_2(a_3-a_1)+b_3(a_1-a_2)=0$ is fulfilled.

We study non-linear oscillations of a nearly integrable Hamiltonian system with one and half degrees of freedom in a neighborhood of an equilibrium. We analyse the resonance case of order one. We perform careful analysis of a small finite neighborhood of the equilibrium. We show that in the case considered the equilibrium is not stable, however, this instability is soft, i.e. trajectories of the system starting near the equilibrium remain close to it for an infinite period of time. We discuss also the effect of separatrices splitting occurring in the system. We apply our theory to study the motion of a particle in a field of waves packet.