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Third and fourth order resonances in Hamiltonian systems

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Abstract

The stability of the origin of an autonomous Hamiltonian system is investigated when the system possesses a third or fourth-order resonance.H 2, the quadratic part ofH isH 2=∑n i=1 ω i J i and the resonance condition is ∑n i=1 k i ω i where thek ⩾ 0,i = 1, 2, ...,n are the natural or fundamental frequencies. It is shown that the only case in which the origin can be unstable is ifk i≥0,i=1,2,...,n. The condition for instability is then given in terms of the coefficients of the higher order terms in the Hamiltonian. The transfer of energy between modes is also investigated when a near-resonant condition exists.

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Authors and Affiliations

  1. Dept. of Theoretical and Applied Mechanics, Cornell University, Ithaca, N.Y., U.S.A.

    Kyle T. Alfriend & David L. Richardson

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  1. Kyle T. Alfriend
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  2. David L. Richardson
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Alfriend, K.T., Richardson, D.L. Third and fourth order resonances in Hamiltonian systems. Celestial Mechanics 7, 408–420 (1973). https://doi.org/10.1007/BF01227507

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  • DOI: https://doi.org/10.1007/BF01227507

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