IOP PUBLISHING NONLINEARITY Nonlinearity 21 (2008) 1759–1811 doi:10.1088/0951-7715/21/8/005 Kolmogorov–Arnold–Moser aspects of the periodic Hamiltonian Hopf bifurcation Merc` e Oll´ e, Juan R Pacha and Jordi Villanueva Departament de Matem` atica Aplicada I, Universitat Polit` ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain E-mail: merce.olle@upc.edu, juan.ramon.pacha@upc.edu and jordi.villanueva@upc.edu Received 25 October 2007, in final form 22 May 2008 Published 26 June 2008 Online at stacks.iop.org/Non/21/1759 Recommended by A Chenciner Abstract In this work we consider a 1 : −1 non-semi-simple resonant periodic orbit of a three degrees of freedom real analytic Hamiltonian system. From the formal analysis of the normal form, we prove the branching off of a two-parameter family of two-dimensional invariant tori of the normalized system, whose normal behaviour depends intrinsically on the coefficients of its low-order terms. Thus, only elliptic or elliptic together with parabolic and hyperbolic tori may detach from the resonant periodic orbit. Both patterns are mentioned in the literature as the direct and inverse, respectively, periodic Hopf bifurcation. In this paper we focus on the direct case, which has many applications in several fields of science. Our target is to prove, in the framework of Kolmogorov– Arnold–Moser (KAM) theory, the persistence of most of the (normally) elliptic tori of the normal form, when the whole Hamiltonian is taken into account, and to give a very precise characterization of the parameters labelling them, which can be selected with a very clear dynamical meaning. Furthermore, we give sharp quantitative estimates on the ‘density’ of surviving tori, when the distance to the resonant periodic orbit goes to zero, and show that the four-dimensional invariant Cantor manifold holding them admits a Whitney-C ∞ extension. Due to the strong degeneracy of the problem, some standard KAM methods for elliptic low-dimensional tori of Hamiltonian systems do not apply directly, so one needs to properly suit these techniques to the context. Mathematics Subject Classification: 37J20, 37J40 1. Introduction This paper is related to the existence of quasiperiodic solutions linked to a Hopf bifurcation scenario in the Hamiltonian context. In its simpler formulation, we shall consider a real 0951-7715/08/081759+53$30.00 © 2008 IOP Publishing Ltd and London Mathematical Society Printed in the UK 1759