On the Hamiltonian Hopf bifurcations in the 3D Hénon-Heiles family

An axially symmetric perturbed isotropic harmonic oscillator undergoes several bifurcations as the parameter λ adjusting the relative strength of the two terms in the cubic potential is varied. We show that three of these bifurcations are Hamiltonian Hopf bifurcations. To this end we analyse an appropriately chosen normal form. It turns out that the linear behaviour is not that of a typical Hamiltonian Hopf bifurcation as the eigen-values completely vanish at the bifurcation. However, the nonlinear structure is that of a Hamiltonian Hopf bifurcation. The result is obtained by formulating geometric criteria involving the normalized Hamiltonian and the reduced phase space. © 2002 Plenum Publishing Corporation.