An Analytical Study in Coupled Map Lattices of Synchronized States and   Travelling Waves, and of their Period-Doubling Cascades
  Several theorems are demonstrated that determine the sufficient conditions for the existence of synchronized states (periodical and chaotic) and also of travelling waves in a CML. Also are analytically proven the existence of period-doubling cascades for the mentioned patterns. The temporal state of any oscillators are completely characterized. The given results are valid for a number of arbitrary oscillators whose individual dynamics is ruled by an arbitrary C^{2} function. 
M. Dolores Sotelo Herrera and Jesus San Martin