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

Abstract

A symplecticity-preserving RG analysis is carried out to study a resonance structure near an elliptic fixed point of a proto-type symplectic map in two dimensions. Through analyzing fixed points of a reduced RG map, a topology of the resonance structure such as a chain of resonant islands can be determined analytically. An application of this analysis to the Hénon map is also presented.

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

“…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. In the paper [2], it was predicted and confirmed that the island structures appear due to the PB resonance near some rational tori.…”

“…(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]. Here, we investigate the map (4) in the more general setting, taking note of the fact that the map (4) includes the integral map of the Suris' type [8].…”

“…In the case k = 3, the resonant secular term dominates over the term of nonlinear frequency shift. For k P 5, the term of nonlinear frequency shift (10) is a dominant nonlinear term while the resonant secular term (11) yields a small but important contribution to the formation of a resonant island chain [2]. Hereafter, we focus our attention on the marginal case k = 4, where the resonant secular term is comparable to the term of nonlinear frequency shift.…”

“…1(b), respectively. viewpoint of integrability of the original map (2). Various integrable maps of the radially twist type (2) are presented by Suris [8].…”

“…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.…”