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

We study period doubling in a symmetric four-dimensional volume-preserving quadratic map, i.e., two symmetrically coupled two-dimensional area-preserving Hénon maps. We must vary two parameters and thus obtain two Feigenbaum constants, ${\delta }_1$ and ${\delta }_2$. 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, ${\delta }_1$ and ${\delta }_2$. We find ${\delta }_2$=+4.000. . .,-2.000. . .,-4.404. . ., respectively, for the three classes. These ${\delta }_2$ values are also recovered here from some approximate (numerical) renormalization scheme. The ${\delta }_1$ is, in all cases, the same as in two-dimensional area-preserving maps, ${\delta }_1$=8.721. . . . The ${\delta }_2$=-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.