Period doubling in coupled maps

Using a renormalization method, we study the critical behavior of period doubling in two coupled one-dimensional (1D) maps. We find three kinds of fixed maps of the period-doubling renormalization operator. Each of the fixed maps has a common relevant eigenvalue associated with scaling of the nonlinearity parameter of the uncoupled 1D map. However, the relevant ``coupling'' eigenvalues associated with scaling of the coupling parameter vary depending on the kind of fixed maps. The values of relevant eigenvalues agree well with those of the parameter scaling factors obtained by a direct numerical method. The renormalization results of two coupled maps are also extended to many coupled maps with a global coupling, in which each map is coupled to all the other maps with equal strength.