Third and fourth order resonances in Hamiltonian systems

The stability of the origin of an autonomous Hamiltonian system is investigated when the system possesses a third or fourth-order resonance. H2, the quadratic part of H is H2=ni=1iJi and the resonance condition is ni=1kii where the k 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 if ki0, 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. 1973 D. Reidel Publishing Company.

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