  We study the stability of the fixed-point solution of an array of mutually
coupled logistic maps, focusing on the influence of the delay times,
$\tau_{ij}$, of the interaction between the $i$th and $j$th maps. Two of us
recently reported [Phys. Rev. Lett. {\bf 94}, 134102 (2005)] that if
$\tau_{ij}$ are random enough the array synchronizes in a spatially homogeneous
steady state. Here we study this behavior by comparing the dynamics of a map of
an array of $N$ delayed-coupled maps with the dynamics of a map with $N$
self-feedback delayed loops. If $N$ is sufficiently large, the dynamics of a
map of the array is similar to the dynamics of a map with self-feedback loops
with the same delay times. Several delayed loops stabilize the fixed point,
when the delays are not the same; however, the distribution of delays plays a
key role: if the delays are all odd a periodic orbit (and not the fixed point)
is stabilized. We present a linear stability analysis and apply some
mathematical theorems that explain the numerical results.
