Critical behavior of period doubling in coupled area-preserving maps

We study the critical behavior of period doublings in $N$ symmetrically coupled area-preserving maps for many-coupled cases with $N>3$. It is found that the critical scaling behaviors depend on the range of coupling interaction. In the extreme long-range case of global coupling, in which each area-preserving map is coupled to all the other area-preserving maps with equal strength, there exist an infinite number of bifurcation routes in the parameter plane, each of which ends at a critical point. The critical behaviors, which vary depending on the type of bifurcation routes, are the same as those for the previously-studied small $N$ cases $(N=2,3)$, independently of $N$. However, for any other non-global coupling cases of shorter range couplings, there remains only one bifurcation route ending at the zero-coupling critical point, at which the $N$ area-preserving maps become uncoupled, The critical behavior at the zero-coupling point is also the same as that for the small $N$ cases $(N=2,3)$, independently of the coupling range.