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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, δ 1 and δ 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-4 q 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, δ 1 and δ 2 . We find δ 2 =+4.000. . ., -2.000. . . , -4.404. . . , respectively, for the three classes. These δ 2 values are also recovered here from some approximate (numerical) renormalization scheme. The δ 1 is, in all cases, the same as in two-dimensional area-preserving maps, δ 1 =8.721. . . . The δ 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.
Physical Review A – American Physical Society (APS)
Published: Feb 15, 1987
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