On the periodic motions of a~Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a~double three-order resonance
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 26 (2016) no. 3, pp. 418-438
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The paper considers the motion of a near-autonomous time-periodic two-degree-of-freedom Hamiltonian system in a neighborhood of trivial equilibrium being stable in the linear approximation. The third-order double resonance (fundamental and Raman) is assumed to occur in the system, at that Raman resonance can be strong or weak. In both cases the equilibrium considered is unstable in a full nonlinear system. Normalization of Hamiltonians of the perturbed motion is carried out in the terms up to the fourth order with respect to disturbance, taking into account the existing resonances. The problem of the existence and number of equilibrium positions of the corresponding approximate (model) systems is solved, and sufficient and necessary conditions for their stability are obtained. By Poincare's small parameter method, periodic motions of the initial full systems generated from the equilibrium positions of the model systems are constructed. The question of their stability in the linear approximation is solved. In particular, the conditions of the existence of stable (in the linear approximation) periodic motions in a small neighborhood of the unstable trivial equilibrium are obtained.
Keywords:
Hamiltonian system, multiple resonance, stability, periodic motion.
@article{VUU_2016_26_3_a9,
author = {A. I. Safonov and O. V. Kholostova},
title = {On the periodic motions of {a~Hamiltonian} system in the neighborhood of unstable equilibrium in the presence of a~double three-order resonance},
journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
pages = {418--438},
publisher = {mathdoc},
volume = {26},
number = {3},
year = {2016},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VUU_2016_26_3_a9/}
}
TY - JOUR AU - A. I. Safonov AU - O. V. Kholostova TI - On the periodic motions of a~Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a~double three-order resonance JO - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki PY - 2016 SP - 418 EP - 438 VL - 26 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/VUU_2016_26_3_a9/ LA - ru ID - VUU_2016_26_3_a9 ER -
%0 Journal Article %A A. I. Safonov %A O. V. Kholostova %T On the periodic motions of a~Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a~double three-order resonance %J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki %D 2016 %P 418-438 %V 26 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/item/VUU_2016_26_3_a9/ %G ru %F VUU_2016_26_3_a9
A. I. Safonov; O. V. Kholostova. On the periodic motions of a~Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a~double three-order resonance. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 26 (2016) no. 3, pp. 418-438. http://geodesic.mathdoc.fr/item/VUU_2016_26_3_a9/