Renormalization group reduction of the Hénon map and application to the transverse betatron motion in cyclic accelerators

Abstract

Abstract. The renormalization group (RG) method is applied to the study of discrete dynamical systems. As a particular example, the Hénon map is considered as being applied to describe the transverse betatron oscillations in a cyclic accelerator or storage ring possessing a FODO-cell structure with a single thin sextupole. A powerful RG method is developed that is valid correct to fourth order in the perturbation amplitude, and a technique for resolving the resonance structure of the Hénon map is also presented. This calculation represents an application of the RG method to the study of discrete dynamical systems in a unified manner capable of reducing the dynamics of the system both far from and close to resonances, thus preserving the symplectic symmetry of the original map. Contents

Renormalization Group as a Probe for Dynamical Systems

“…The Boltzmann equation has been derived as a RG equation [116]. It has also found applications in accelerator physics [117,118], astrophysics [119], fluid dynamics [120,121], material sciences [122,123] and so on.…”

Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

“…After their works, many studies of the RG method have been done [10][11][12][13]16,21,22,[26][27][28][29][30][31][32]35,36,42,45,46,48,49,53,[55][56][57][58][59]. Kunihiro [28,29] interpreted an approximate solution obtained by the RG method as the envelope of a family of regular perturbation solutions.…”

Renormalization Analysis of Resonance Structure in a 2-D Symplectic Map

“…A map (4) with a m = 0 for m > 2 seems to be rather special but has been extensively studied as the Hénon map[7] [8]. Let us apply results in the preceding sections to the Hénon map and compare them with numerical calculations.…”