The Hamiltonian Hopf bifurcation: an elementary perturbative approach

The Hamiltonian Hopf bifurcation is briefly introduced. Its occurrence in a problem involving a Rydberg electron in a rotating electric field is point ed out by way of illustration. Starting from a set of canonical variables i n terms of which the quadratic part of the Hamiltonian assumes a simple sta ndard form, we go over to polar co-ordinates that enable us in a natural wa y to identify 'fast' and 'slow' variables in the problem. The role of terms of higher degree is then analysed in a perturbative approach, employing th e technique of averaging over the fast variables. The terms necessary to de scribe all the essential features of the bifurcation are identified, arrivi ng at a simple separable Hamiltonian incorporating the bifurcation characte ristics involving families of periodic orbits. It is shown that the bifurca tion can essentially be of two types and that one of those involves an inte rsting 'secondary bifurcation' phenomenon, thereby confirming results of ea rlier analyses of the problem. The simple approach presented here additiona lly allows us to compute tori around the periodic orbits. We briefly illust rate our method by referring to the 'spinning orthogonal double pendulum'. Remarks on the significance of the Hamiltonian Hopf bifurcation in the cont ext of KAM theory and on the quantisation of the bifurcation are included. (C) 2001 Elsevier Science Ltd. All rights reserved.