Global Synchronization in Translation Invariant Coupled Map Lattices

A sufficient condition for global synchronization in coupled map lattices (CML) with translation invariant coupling and arbitrary individual map is proved. As in [Jost & Joy, 2001] where CML with reflection invariant couplings are considered, the condition only involves the linearized dynamics in the diagonal, namely for all points in the diagonal, the derivative must be contractive in all transverse directions. In addition to this result, a (weaker) condition that ensures the CML attractor to be composed of either 2-periodic or constant configurations, is also obtained.