Synchronized family dynamics in globally coupled maps

The dynamics of a globally coupled, logistic map lattice is explored over a parameter plane consisting of the coupling strength, epsilon, and the map parameter, a. By considering simple periodic orbits of relatively small lat tices, and then an extensive set of initial-value calculations, the phenome nology of solutions over the parameter plane is broadly classified. The lat tice possesses many stable solutions, except for sufficiently large couplin g strengths, where the lattice elements always synchronize, and for small m ap parameter, where only simple fixed points are found. For smaller epsilon and larger a, there is a portion of the parameter plane in which chaotic, asynchronous lattices are found. Over much of the parameter plane, lattices converge to states in which the maps are partitioned into a number of sync hronized families. The dynamics and stability of two-family states (solutio ns partitioned into two families) are explored in detail. (C) 1999 American Institute of Physics. [S1054-1500(99)01503-7].