Period p-tuplings in coupled maps

We study the critical behavior (CB) of all period $p$-tuplings $(p \!=\!2,3,4,\dots)$ in $N$ $(N \!=\! 2,3,4,\dots)$ symmetrically coupled one-dimensional maps. We first investigate the CB for the $N=2$ case of two coupled maps, using a renormalization method. Three (five) kinds of fixed points of the renormalization transformation and their relevant ``coupling eigenvalues'' associated with coupling perturbations are found in the case of even (odd) $p$. We next study the CB for the linear- and nonlinear-coupling cases (a coupling is called linear or nonlinear according to its leading term), and confirm the renormalization results. Both the structure of the critical set (set of the critical points) and the CB vary according as the coupling is linear or nonlinear. Finally, the results of the two coupled maps are extended to many coupled maps with $N \geq 3$, in which the CB depends on the range of coupling.