Nonlinear analysis of perturbed ion traps V. Lanchares † , J. Palaci´ an ‡ A. I. Pascual † , J. P. Salas ∗ and P. Yanguas ‡ †Departamento de Matem´aticas y Computaci´on. ∗ ´ Area de F´ ısica Aplicada. Universidad de La Rioja. E – 26 004 Logro˜ no. Spain. ‡Departamento de Matem´atica e Inform´ atica. Universidad P´ ublica de Navarra. E – 31 006 Pamplona. Spain. Monograf´ ıas de la Real Academia de Ciencias de Zaragoza. 22: 37–44, (2003). Abstract This paper presents an analytical study of a perturbed ion trap. Various tech- niques are combined in order to extract information about the evolution of the sys- tem. The problem is modelled by an axially–symmetric–three–degree–of–freedom Hamiltonian. Normalization plus reduction lead to an integrable system whose flow is analyzed. Finally a qualitative relationship between the flow associated to the integrable system and the one attached to the original Hamiltonian is established. For this purpose we use estimations of the error in the normalization, Poincar´ e sur- faces of section and KAM theory. Key words and expressions: Hamiltonian, perturbation, trap, normalization, reduction, bifurcation. AMS (MOS) Subject Classification: 37J40, 37J15, 37J20, 70K65. 1 Introduction Since the beginning of last century, the effect of the application of external fields to atoms has played a crucial role in the development of atomic physics. In particular, the application of static electric and magnetic fields to create trapping phenomena is a remarkable feature. When the trapped particle is an ion, lab experiments are used to perform very precise spectroscopic measurements and to construct accurate atomic clocks. In this paper we focus on one of these experiments: the Penning trap [7, 5]. Briefly described, the Penning trap represents a three–dimensional trapping of a charge or ion due to an axially–symmetric (“perfect”) quadrupole electric field plus a static magnetic field. The perfect quadrupole electric potential is achieved by means of a set of 37