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Interaction of Lower and Higher Order Hamiltonian Resonances

Ferdinand Verhulst

Mathematisch Instituut, P. O. Box 80.010, 3508TA Utrecht, Netherlands

https://doi.org/10.1142/S0218127418500979Cited by:3 (Source: Crossref)

Abstract

The tools of normal forms and recurrence are used to analyze the interaction of low and higher order resonances in Hamiltonian systems. The resonance zones where the short-periodic solutions of the low order resonances exist are characterized by small variations of the corresponding actions that match the variations of the higher order resonance; this yields cases of embedded double resonance. The resulting interaction produces periodic solutions that in some cases destabilize a resonance zone. Applications are given to the three dof $1 : 1 : 4$ $1 : 1 : 4$ resonance and to periodic FPU-chains producing unexpected nonlinear stability results and quasi-trapping phenomena.

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