New Feigenbaum constants for four-dimensional volume-preserving symmetric maps.

Abstract

We study period doubling in a symmetric four-dimensional volume-preserving quadratic map, i.e., two symmetrically coupled two-dimensional area-preserving H\'enon maps. We must vary two parameters and thus obtain two Feigenbaum constants, ${\ensuremath{\delta}}_{1}$ and ${\ensuremath{\delta}}_{2}$. It is a very important point that for each region of stability (belonging to some period-q orbit) in this parameter plane we find two regions of stability for the period-2q orbit, four regions for the period-4q orbit, and so on. Hence we have an infinite number of stability regions and infinities of bifurcation ``paths'' through these regions. Almost all self-similar bifurcation paths fall into one of three possible ``universality classes,'' i.e., each class is characterized by its own two Feigenbaum constants, ${\ensuremath{\delta}}_{1}$ and ${\ensuremath{\delta}}_{2}$. We find ${\ensuremath{\delta}}_{2}$=+4.000. . .,-2.000. . .,-4.404. . ., respectively, for the three classes. These ${\ensuremath{\delta}}_{2}$ values are also recovered here from some approximate (numerical) renormalization scheme. The ${\ensuremath{\delta}}_{1}$ is, in all cases, the same as in two-dimensional area-preserving maps, ${\ensuremath{\delta}}_{1}$=8.721. . . . The ${\ensuremath{\delta}}_{2}$=-15.1. . ., reported in an earlier paper [J. M. Mao, I. Satija, and B. Hu, Phys. Rev. A 32, 1927 (1985)], applies to only two exceptional paths.