Non-linear oscillations of a Hamiltonian system with 2:1 resonance |
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| Institution: | 1. Department of Emergency Medicine, Cooper Medical School of Rowan University, Camden, NJ;2. Department of Emergency Medicine, University of California at San Francisco School of Medicine, San Francisco, CA;3. Division of Emergency Medicine, Department of Pediatrics, Children’s Hospital of Michigan, Detroit, MI;4. College of Medicine, Central Michigan University, Mount Pleasant, MI;5. Department of Pediatric Emergency Medicine, Children’s Hospital of New Orleans, New Orleans, LA;6. Division of Emergency Medicine, Department of Surgery, Duke University Hospital, Durham, NC;7. Division of Pediatric Emergency Medicine, Department of Emergency Medicine, Massachusetts General Hospital, Boston MA;8. Division of Pediatric Emergency Medicine, Department of Emergency Medicine, Cooper Medical School of Rowan University, Camden, NJ;9. Division of Emergency Medicine, Children’s Hospital of Philadelphia, Philadelphia, PA;10. Section of Emergency Medicine, Department of Pediatrics, Texas Children’s Hospital/Baylor College of Medicine, Houston, TX;11. Department of Emergency Medicine, Jackson Memorial Hospital/Holtz Children’s Hospital, Miami, FL;12. Department of Pediatric Emergency Medicine, St. Christopher’s Hospital for Children, Philadelphia, PA;1. Hebei Key Lab of Power Plant Flue Gas Multi-Pollutants Control, Department of Environmental Science and Engineering, North China Electric Power University, Baoding 071003, PR China;2. MOE Key Laboratory of Resources and Environmental Systems Optimization, College of Environmental Science and Engineering, North China Electric Power University, Beijing 102206, PR China |
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| Abstract: | Non-linear oscillations of an autonomous Hamiltonian system with two degrees of freedom in the neighbourhood of a stable equilibrium are considered. It is assumed that the frequency ratio of the linear oscillations is close to or equal to two, and that the Hamiltonian is sign-definite in the neighbourhood of the equilibrium. A solution is presented to the problem of the orbital stability of periodic motions emanating from the equilibrium position. Conditionally periodic motions of an approximate system are analysed taking into account terms of order up to and including three in the normalized Hamiltonian. The KAM theory is used to consider the problem of maintaining these motions taking into account fourth- and higher-order terms in the series expansion of the Hamiltonian in a sufficiently small neighbourhood of the equilibrium. The results are used to investigate non-linear oscillations of an elastic pendulum. |
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