Steady-state stabilization due to random delays in maps with self-feedback loops and in globally delayed-coupled maps

Abstract

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, , of the interaction between the and maps. Two of us recently reported [Phys. Rev. Lett. 94, 134102 (2005)] that if 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 delayed-coupled maps with the dynamics of a map with self-feedback delayed loops. If 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.