The 1:3:4 resonance Dynamics near a resonant equilibrium

Abstract

We study the dynamics near a local extremum of a Hamiltonian function, for which the frequencies of the linearization are in 1:3:4 resonance. The 1:3:4 resonance is a genuine second order resonance, for which the truncated second normal form is expected to no longer be integrable, as opposed to the integrable truncated first normal form.

The dynamics of the first normal form are studied using singular reduction. We consider a detuning and the addition of the quartic self-interaction terms; Hamiltonian Hopf bifurcations are observed under variation of internal parameters. We present stability results on the normal modes for the first normal forms of various families of resonances. We consider the singular reduction, with respect to the periodic flow of the quadratic Hamiltonian, and we analyse the dynamics of the second normal form. Lastly, we study the dynamics of the indefinite 1:3:-4 resonance.