Regularized Renormalization Group Reduction of Symplectic Maps

Abstract

By means of the perturbative renormalization group method, we study a long-time behaviour of some symplectic discrete maps near elliptic and hyperbolic fixed points. It is shown that a naive renormalization group (RG) map breaks the symplectic symmetry and fails to describe a long-time behaviour. In order to preserve the symplectic symmetry, we present a regularization procedure, which gives a regularized symplectic RG map describing an approximate long-time behaviour succesfully. *

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

“…Moreover, it was shown that, using the implicit amplitude-phase map ansatz introduced in section 3, one can extract a renormalized symplectic map and thus recover the symplectic structure of the original Hénon map. This technique yields results similar to the ones obtained by means of the regularization procedure [15] and [17] proposed previously. The present calculation represents a successful application of the powerful RG method to the study of discrete dynamical systems far from and close to resonances in a unified manner.…”

“…To recover the symplectic symmetry, we regularize [15] the naive RG map (2.35) by noting that the coefficient in the square brackets multiplyingÃ(n) can be exponentiated as…”

“…The recently developed renormalization group (RG) method has been successfully applied to both continuous dynamical systems [12]- [14] and maps [15,16] that are of general interest in the physics of accelerators and beams. In a recent paper by Goto et al [17], a symplectic map chain was investigated by means of a new form of the regularized RG method previously introduced in [15]. This new form of the regularization procedure (the symplectic integration method) is similar in spirit to our method of extraction of a symplectic RG map utilizing the implicit amplitude-phase ansatz described in section 3.…”

Perturbative renormalization analysis of the Poincaré–Birkhoff resonance in 2-D symplectic map

“…If among the coefficients of the nonlinear terms, only a 3 is non-vanishing, then Eq. (4) takes the form of the double-well type map [4,7]. Some resonant structures of these two maps were studied near the elliptic fixed point by means of the renormalization method [2].…”

“…However, there has been not yet such a systematic method for a symplectic map. Recently, a new direct method (called the renormalization method) has been developed to analyze a long-time behavior of orbits of a two-dimensional symplectic map [2][3][4]. Since this direct method is a discrete version of the perturbative renormalization group method for a flow system [5], we call it the renormalization method or the R method here.…”