Hamiltonian Hopf bifurcations near a chaotic Hamiltonian resonance

Abstract

The 1:2:3 Hamiltonian resonance is one of the four genuine first order resonances which is non-integrable. For this resonance, chaotic behavior of the normal form has been shown due to the existence of a transverse homoclinic orbit on the energy manifold. Considering the detuning parameters, in the mirror symmetric cases of the Poisson manifold, by a reduction theory and computing the classical normal form of non-degenerate Hamiltonian Hopf bifurcation, we show there are just non-degenerate Hamiltonian Hopf bifurcations. However in the general case by considering some case studies and specially of FPU (Fermi–Pasta–Ulam) chains for a fiber passaging 1:2:3 resonance, we may deal with complex dynamics and specially Hamiltonian Hopf bifurcation in a complicated region. Actually, we can see some special Krein collisions in a complex region.