Abstract
The study of the dynamics of nonautonomous mappings or of parametric dependent dynamical systems has been considered in previous works[1,2,3], where a generalization of some results of autonomous dynamical systems is given under certain conditions. In this paper we present the generalization of Birkhoff normal forms[4,5] and of the KAM theory for the construction of quasi-periodic tori[6,7,8,9,10] to the dynamics of nonautonomous symplectic maps which depend on time in a periodic or quasi-periodic way. In particular we are interested in the stability properties of an elliptic fixed point of an analytic periodically time-dependent diffeomorphism, which is globally symplectic. The interest on this problem arises from accelerator physics: the betatronic motion in a circular particle accelerator (i.e. the motion on a plane perpendicular to the reference orbit) is described by a symplectic map <Emphasis FontCategory=“NonProportional”>M</Emphasis> in ℝ4 (the one-turn map) with an elliptic fixed point at the origin corresponding to the reference orbit[11,12]. The multipolar magnetic fields, which are caused by the superconducting magnets[13], introduce nonlinear terms in the one-turn map so that it is necessary to study the stability properties of the origin in order to have previsions on the beam’s lifetime.
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Bazzani, A. (1994). Kam Tori for Modulated Symplectic Maps. In: Kuksin, S., Lazutkin, V., Pöschel, J. (eds) Seminar on Dynamical Systems. Progress in Nonlinear Differential Equations and Their Applications, vol 12. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7515-8_11
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DOI: https://doi.org/10.1007/978-3-0348-7515-8_11
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