Stability of triangular libration points in a planar restricted elliptic three body problem in cases of double resonances

Kholostova, O.V.

doi:10.1016/j.ijnonlinmec.2014.11.005

Cited by 6 publications

(7 citation statements)

References 4 publications

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“…Generally, a small parametric excitation can produce a large response, and the system loses its stability when the frequency of the parametric excitation is close to double of the system's natural frequency, which is so called principal parametric resonance. Following the interesting feature of this system, the stability analysis for coupled and parametric excited system like BCP model can be solved directly with application of the perturbation methods, for example, harmonic balance method [24], multiple scale method [25][26][27][28][29]. The representative work was demonstrated by Alfriend and Rand [30] who applied a general perturbation technique to study the stability of infinitesimal motions in the non-autonomous elliptic RTBP.…”

Section: Candiceqyj@163commentioning

confidence: 99%

Parametric stability analysis for planar bicircular restricted four-body problem

Qian

Yang

et al. 2018

Astrodyn

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Section: Candiceqyj@163commentioning

confidence: 99%

Parametric stability analysis for planar bicircular restricted four-body problem

Qian

Yang

et al. 2018

Astrodyn

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“…При указанных значениях параметров нормализованный до членов четвертой степени гамильтониан возмущенного движения имеет вид, аналогичный (2.5) (за независимую переменную принята истинная аномалия ν) [18]: Параметры (4.2) лежат в области (3.11), откуда следует, что все движения модельной системы ограниченны.…”

Section: область ограниченности движений в задаче о треугольных точкаunclassified

“…Воспользовавшись результатами раздела 3 (см. также [18]), найдем, что на отрицательных и нулевом уровнях энергии величины r 1 и r 2 меняются в пределах интервалов…”

Section: область ограниченности движений в задаче о треугольных точкаunclassified

“…Показано, что существуют области параметров (коэффициентов нормализованного гамильтониана), в которых все движения приближенной (модельной) системы ограничены, получена оценка области ограниченности движений. Этот результат является обобщением проведенного ранее исследования задачи о треугольных точках либрации плоской эллиптической ограниченной задачи трех тел в случае рассматриваемого кратного резонанса [18]. В работе также выявлено, что взаимодействие двух резонансов, каждый из которых не приводит к неустойчивости, оказывает возмущающее воздействие на движения модельной системы в пределах области ограниченности.…”

Section: Introductionunclassified

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The interaction of resonances of the third and fourth orders in a Hamiltonian two-degree-of-freedom system

Kholostova¹

2015

Nelin. Dinam.

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“…The problem of the stability of an autonomous Hamiltonian system in the degenerate (transcendental) resonant case was studied in [24]. Multiple resonances, different from the 1 : 1 : 1 resonance, in a nonautonomous Hamiltonian system with two degrees of freedom were investigated in the recent series of publications [25][26][27][28][29][30][31][32]. General theoretical results obtained in these works were applied in investigating the motion of a satellite, a rigid body, relative to the center of mass and in analyzing the stability of Lagrangian solutions to the planar bounded elliptic three-body problem.…”

Section: Introductionmentioning

confidence: 99%

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

Markeev¹,

Chekhovskaya²

2020

Nelin. Dinam.

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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).

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Stability of triangular libration points in a planar restricted elliptic three body problem in cases of double resonances

Cited by 6 publications

References 4 publications

Parametric stability analysis for planar bicircular restricted four-body problem

Parametric stability analysis for planar bicircular restricted four-body problem

The interaction of resonances of the third and fourth orders in a Hamiltonian two-degree-of-freedom system

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

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