Long transients and cluster size in globally coupled maps

– We analyze the asymptotic states in the partially ordered phase of a system of globally coupled logistic maps. We confirm that, regardless of initial conditions, these states consist of a few clusters, and they properly belong in the ordered phase of these systems. The transient times necessary to reach the asymptotic states can be very long, especially very near the transition line separating the ordered and the coherent phases. We find that, where two clusters form, the distribution of their sizes corresponds to windows of regular or narrow-band chaotic behavior in the bifurcation diagram of a system of two degrees of freedom that describes the motion of two clusters, where the size of one cluster acts as a bifurcation parameter. Systems formed by globally coupled logistic maps [1–3] are a paradigm of the rich phenomenology arising when opposing tendencies compete: the nonlinear dynamics of the maps, which in the chaotic regime tends to separate the orbits of different elements, and the coupling, that tends to synchronize them. The synchronization of dynamical elements from an undifferentiated state makes these systems particularly interesting in many interdisciplinary applications, such as network of optical elements, networks of Josephson junctions, ecological systems and social behavior [4–6]. The equations describing the evolution of a system with N globally coupled elements are: xi(t+ 1) = (1 − ǫ)f [xi(t)] + ǫ N ∑