On equilibrium positions bifurcations of Hamiltonian system in cases of double combined third order resonance

Theoretical mechanics


Holostova O. V.1*, Safonov A. I.2**

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. InfoSistem-35, 16, 3-Mytischinskaya str., bld. 37, Moscow, 129626, Russia

*e-mail: kholostova_o@mail.ru
**e-mail: lexafonov@mail.ru


Motion of a near-autonomous time-periodic two-degree-of-freedom Hamiltonian system is considered near the trivial equilibrium position, stable in the linear approximation. The system is assumed to realize simultaneously two Raman third-order resonances being strong and weak. In this case, the frequencies of small oscillations are related also by the fourth-order resonant ratio. In the complete nonlinear system, this equilibrium position is unstable. By the perturbation theory methods, normalization of the Hamiltonian of the perturbed motion is performed in the members up to the fourth order inclusively with respect to disturbance, and account for the existing resonances. On the assumption that the summand coefficients related to the fourth-order resonance are small, the approximate (model) system, dependent on the three parameters, was considered. The issue of existence and number of equilibrium positions of the model system was solved by analytical and graphics methods. The domains, where the number of equilibrium positions of the model system can vary from zero to five were identified in the space of parameters of the problem.

As an application, the problem of motion of a dynamically symmetric satellite (modeled by a rigid body) about the center of mass in the central Newtonian gravitational field on the orbit with small eccentricity was considered. The satellite motion was studied near its periodic motion, emerging from the hyperboloid precession on a circular orbit. For parameter values, corresponding to the multiple Raman third-order resonance, the coefficients of the normal forms of the model system are found. It was shown, that there are three equilibrium points in this system, two of which are unstable and the third stable in the linear approximation.


Hamiltonian system, multiple Raman resonance, dynamically symmetric satellite, hyperboloid precession


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