Non-linear oscillations of a Hamiltonian system with two degrees of freedom with 2:1 resonance |
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| Authors: | OV Kholostova |
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| Institution: | 1. Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC, USA;2. Department of Civil and Environmental Engineering, University of Delaware, Newark, DE, USA;3. Department of Mechanical Engineering, Faculty of Technical Sciences, Novi Sad, Trg D. Obradovica 6, Serbia;1. School of Mathematical Sciences, Beihang University (BUAA), Beijing 100191, PR China;2. LAGA, UMR 7539, Institut Galilée, Université Sorbonne Paris Nord, 93430, Villetaneuse, France;1. Department of Civil Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran;2. Department of Civil and Structural Engineering, The University of Sheffield, Sheffield, UK;1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, PR China;2. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241000, Anhui, PR China;1. Zhongtai Securities Institute for Financial Studies, Shandong University, China;2. School of Mathematics, Shandong University, China |
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| Abstract: | The motions of an autonomous Hamiltonian system with two degrees of freedom close to an equilibrium position, stable in the linear approximation, are considered. It is assumed that in this neighbourhood the quadratic part of the Hamiltonian of the system is sign-variable, and the ratio of the frequencies of the linear oscillations are close to or equal to two. It is also assumed that the corresponding resonance terms in the third-degree terms of the Hamiltonian are small. The problem of the existence, bifurcations and orbital stability of the periodic motions of the system near the equilibrium position is solved. Conditionally periodic motions of the system are investigated. An estimate is obtained of the region in which the motions of the system are bounded in the neighbourhood of an unstable equilibrium in the case of exact resonance. The motions of a heavy dynamically symmetrical rigid body with a fixed point in the neighbourhood of its permanent rotations around the vertical for 2:1 resonance are considered as an application. |
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