Period Doubling in Four-Dimensional Volume-Preserving Maps

TL;DR

This paper investigates the detailed scaling behavior of period doubling bifurcations in a four-dimensional volume-preserving map, revealing a new scaling factor and confirming previous renormalization predictions.

Contribution

It introduces a two-term scaling law for period doubling in four-dimensional maps and identifies a new scaling factor associated with coupling parameter scaling.

Findings

Discovered a new scaling factor δ₃ ≈ 1.8505

Confirmed previous renormalization results by Mao and Greene

Extended the scaling law to include coupling effects

Abstract

We numerically study the scaling behavior of period doublings at the zero-coupling critical point in a four-dimensional volume-preserving map consisting of two coupled area-preserving maps. 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 $\delta_3$ $(=1.8505\dots)$ associated with scaling of the coupling parameter, in addition to the previously known scaling factors $\delta_1$ $(=-8.7210\dots)$ and $\delta_2$ $(=-4.4038\dots)$. These numerical results confirm the renormalization results reported by Mao and Greene [Phys. Rev. A {\bf 35}, 3911 (1987)].