On the Motions of One Near-Autonomous Hamiltonian System at a 1:1:1 Resonance

We consider the motions of a near-autonomous 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 three-frequency 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.

Section: General Case Model System Of the First Approximationmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

On Nonlinear Oscillations of a Time-Periodic Hamiltonian System at a 2:1:1 Resonance

Kholostova¹

2022

“…We note that there is also a point α = 1, β = 2 at which equation (8.1) has a double root ω 1 = ω 2 = 1 and the above-mentioned quadratic form reduces to the sum of squares. This case was previously examined in detail in [6].…”

Section: On Nonlinear Oscillations Of a Near-autonomous Hamiltonian Systemmentioning

confidence: 99%

“…A more detailed analysis of the structure of stability and instability regions in a number of cases of multiple parametric resonances was carried out in [4][5][6][7], whereas this paper deals with problems of the existence, number and stability of nonlinear resonant periodic motions of the system and problems of the existence of conditionally periodic motions [6].…”

Section: Introductionmentioning

confidence: 99%

On Nonlinear Oscillations of a Near-Autonomous Hamiltonian System in the Case of Two Identical Integer or Half-Integer Frequencies

Kholostova¹

2021

This paper examines the motion of a time-periodic 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 half-integer 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.

“…In [15,20] the 1 : 1 resonance was considered in the problem of the motion of a satellite, a rigid body, relative to the center of mass in a circular orbit. Nonlinear oscillations of a dynamically symmetric satellite at the 1 : 1 : 1 resonance in an elliptic orbit of small eccentricity were examined in [21][22][23]. The problem of the stability of an autonomous Hamiltonian system in the degenerate (transcendental) resonant case was studied in [24].…”

Section: Introductionmentioning

confidence: 99%

On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance

Markeev¹,

Chekhovskaya²

2020

The points of suspension of two identical pendulums moving in a homogeneous gravitational field are located on a horizontal beam performing harmonic oscillations of small amplitude along a fixed horizontal straight line passing through the points of suspension of the pendulums. The pendulums are connected to each other by a spring of low stiffness. It is assumed that the partial frequency of small oscillations of each pendulum is exactly equal to the frequency of horizontal oscillations of the beam. This implies that a multiple resonance occurs in this problem, when the frequency of external periodic action on the system is equal simultaneously to two its frequencies of small (linear) natural oscillations. This paper solves the nonlinear problem of the existence and stability of periodic motions of pendulums with a period equal to the period of oscillations of the beam. The study uses the classical methods due to Lyapunov and Poincaré, KAM (Kolmogorov, Arnold and Moser) theory, and algorithms of computer algebra. The existence and uniqueness of the periodic motion of pendulums are shown, its analytic representation as a series is obtained, and its stability is investigated. For sufficiently small oscillation amplitudes of the beam, depending on the value of the dimensionless parameter which characterizes the stiffness of the spring connecting the pendulums, the found periodic motion is either Lyapunov unstable or stable for most (in the sense of Lebesgue measure) initial conditions or formally stable (stable in an arbitrarily large, but finite, nonlinear approximation).