Extension of "Renormalization of period doubling in symmetric four-dimensional volume-preserving maps"

We numerically reexamine the scaling behavior of period doublings in fourdimensional volume-preserving maps in order to resolve a discrepancy between numerical results on scaling of the coupling parameter and the approximate renormalization results reported by Mao and Greene [Phys. Rev. A 35, 3911 (1987)]. In order to see the fine structure of period doublings, we extend the simple one-term scaling law to a two-term scaling law. Thus we find a new scaling factor associated with coupling and confirm the approximate renormalization results. PACS numbers: 05.45.+b, 03.20.+i, 05.70.Jk Typeset using REVTEX 1 Universal scaling behavior of period doubling has been found in area-preserving maps [1–7]. As a nonlinearity parameter is varied, an initially stable periodic orbit may lose its stability and give rise to the birth of a stable period-doubled orbit. An infinite sequence of such bifurcations accumulates at a finite parameter value and exhibits a universal limiting behavior. However these limiting scaling behaviors are different from those for the onedimensional dissipative case [8]. An interesting question is whether the scaling results of area-preserving maps carry over higher-dimensional volume-preserving maps. Thus period doubling in four-dimensional (4D) volume-preserving maps has been much studied in recent years [7,9–13]. It has been found in Refs. [11–13] that the critical scaling behaviors of period doublings for two symmetrically coupled area-preserving maps are much richer than those for the uncoupled area-preserving case. There exist an infinite number of critical points in the space of the nonlinearity and coupling parameters. It has been numerically found in [11,12] that the critical behaviors at those critical points are characterized by two scaling factors, δ1 and δ2. The value of δ1 associated with scaling of the nonlinearity parameter is always the same as that of the scaling factor δ (= 8.721 . . .) for the area-preserving maps. However the values of δ2 associated with scaling of the coupling parameter vary depending on the type of bifurcation routes to the critical points. The numerical results [11,12] agree well with an approximate analytic renormalization results obtained by Mao and Greene [13], except for the zero-coupling case in which the two area-preserving maps become uncoupled. Using an approximate renormalization method including truncation, they found three relevant eigenvalues, δ1 = 8.9474, δ2 = −4.4510 and δ3 = 1.8762 for the zero-coupling case [14]. However they believed that the third one δ3 is an artifact of the truncation, because only two relevant eigenvalues δ1 and δ2 could be indentified with the scaling factors numerically found. In this Brief Report we numerically study the critical behavior at the zero-coupling point in two symmetrically coupled area-preserving maps and resolve the discrepancy between the numerical results on the scaling of the coupling parameter and the approximate renormaliza2 tion results for the zero-coupling case. In order to see the fine structure of period doublings, we extend the simple one-term scaling law to a two-term scaling law. Thus we find a new scaling factor δ3 = 1.8505 . . . associated with coupling, in addition to the previously known coupling scaling factor δ2 = −4.4038 . . . . The numerical values of δ2 and δ3 are close to the renormalization results of the relevant coupling eigenvalues δ2 and δ3. Consequently the fixed map governing the critical behavior at the zero-coupling point has two relevant coupling eigenvalues δ2 and δ3 associated with coupling perturbations, unlike the cases of other critical points. Consider a 4D volume-preserving map T consisting of two symmetrically coupled areapreserving Hénon maps [11,12],