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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
L.D. Pustyl’nikhov, The stability of a semitrajectory under transformation of the plane that preserves area, Soviet Math. Dokl. 13, 36–38 (1972)
L.D. Pustyl’nikhov, The existence of a set of positive measure of oscillating motions in a certain problem of dynamics, Soviet Math. Dokl. 13, 94–97 (1972)
H.W. Broer, G.B. Huitema, F. Takens, B.L.J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, Memoirs Am. Math. Soc. 83 n. 421 (1990)
G.D. Birkhoff, Surface transformations and their dynamical applications, Acta Math. 43, 1–119 (1920)
A.D. Brjuno, The analytical forms of differential equations: part I. Trans. Mosc. Math. Soc. 25, 131–288 (1972)
A.N. Kolmogorov, On conservation of conditionally periodic motion for small change in Hamilton’s function, Dokl. Akad. Nauk SSSR 98, 527–530 (1954)
V.I. Arnol’d, Proof of A.N. Kolmogorov theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian Russian, Math. Surveys 18, n.6, 9–36 (1963)
J. Moser, On invariant curves of area-preserving mappings of the annulus, Nachr. Akad. Wiss. Gott. 1, 1–20 (1962)
N.V. Svanidze, Small perturbations of an integrable dynamical system with integral invariant, Proc. Steklov Inst. Math. 2, 127–151 (1981)
M.R. Herman, On the dynamics on lagrangian tori invariant by symplectic diffeomorphisms, Preprint Ecole Polytechnique
A. Dragt, J.M. Finn, Lie Series and Invariant Functions for Analytic Symplectic Maps, J. Math. Phys. 20. 2649 (1979)
A. Bazzani, P. Mazzanti, G. Servizi, G. Turchetti, Normal Forms for Hamiltonian maps and Nonlinear Effects in a Particle Accelerator, 102 B, N. 1, 51–80 (1988)
W. Scandale, G. Turchetti (eds.), Nonlinear Problems in Future Particles Accelerators, World Scientific (1991)
C. Altuna et al., The 1991 Dynamic Aperture Experiment at the CERN SPS, CERN SL 91–43(AP) (1991)
J. Moser, Stabiliätsverhalten kanonischer Differentialgleichungssysteme, Nachr. Akad. Wiss. Göttingen, Math. Phys. K1 IIa, 6, 87–120 (1955)
J. Glimm, Formal Stability of Hamiltonian Systems, Comm. Pure Appl. Math. 17, 507–526 (1964)
A. Giorgilli, L. Galgani, Rigorous estimates for the series expansions of hamiltonian perturbation theory, Cel. Mech. 17 95–112 (1985)
A. Bazzani, S. Marmi, G. Turchetti, Nekhoroshev estimates for isochronous nonresonant symplectic maps, Cel. Mech. 47, 333 (1990)
A. Bazzani, in preparation (1992)
A.I. Neishtadt, Estimates in the Kolmogorov theorem on the conservation of conditionallyperiodic motions, J. Appl. Math. Mech. 48, 58–63 (1982)
V.I. Arnol’d, Instability of dynamical systems with several degrees of freedom, Sov. Math. Dokl. 5, 581–585 (1964)
A. Bazzani, Proprietá di analiticitá delle serie perturbative e stabilitá in dinamica dei fasci, Ph.d Thesis Univ. of Bologna (1992)
C.L. Siegel, J. Moser, Lectures on Celestial Mechanics, Springer-Verlag (1971)
N.N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltoian systems, Russ. Math. Surveys 32, 1–66 (1977)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Basel AG
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7515-8_11
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7517-2
Online ISBN: 978-3-0348-7515-8
eBook Packages: Springer Book Archive