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, ${\tau }_{ij}$, 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.

14 pages, 13 figures, important changes (title changed, discussion, figures, and references added)

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• Discontinuous Attractor Dimension at the Synchronization Transition of Time-Delayed Chaotic Systems
• Complete synchronization equivalence in asynchronous and delayed coupled maps
• Dynamics of delayed-coupled chaotic logistic maps: influence of network topology, connectivity and delay times
• From synchronous to one-time delayed dynamics in coupled maps