COUPLED MAPS ON TREES

We study coupled maps on a Cayley tree, with local (nearest-neighbor) interactions, and with a variety of boundary conditions. The homogeneo us state (where every lattice site has the same value) and the node-sy nchronized state (where sites of a given generation have the same valu e) are both shown to occur for particular values of the parameters and coupling constants. We study the stability of these states and their domains of attraction. Since the number of sites that become synchroni zed is much higher compared to that on a regular lattice, control is e asier to achieve. A general procedure is given to deduce the eigenvalu e spectrum for these states. Perturbations of the synchronized state l ead to different spatiotemporal structures. We find that a mean-field- like treatment is valid on this (effectively infinite dimensional) lat tice.