Coupled maps on trees

TLDR: The homogeneous state and the node-synchronized state are both shown to occur for particular values of the parameters and coupling constants, and a general procedure is given to deduce the eigenvalue spectrum for these states.

Abstract: We study coupled maps on a Cayley tree, with local (nearest-neighbor) interactions, and with a variety of boundary conditions. The homogeneous state (where every lattice site has the same value) and the node-synchronized state (where sites of a given generation have the same value) 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 synchronized is much higher compared to that on a regular lattice, control is easier to achieve. A general procedure is given to deduce the eigenvalue spectrum for these states. Perturbations of the synchronized state lead to different spatiotemporal structures. We find that a mean-field-like treatment is valid on this (effectively infinite dimensional) lattice.