Spatially Coherent States in Fractally Coupled Map Lattices

We study coupled map lattices with a scaling form of connectivity and show that the dynamics of these systems exhibit a transition from spatial disorder to spatially uniform, temporal chaos as the scaling is varied. We numerically investigate the eigenvalue spectrum of the random matrix characterizing fluctuations from spatial uniformity, and find that the spectrum is real, bounded, and has a gap between the largest eigenvalue (corresponding to the uniform solution) and the remaining $N-1\mathrm{}$ eigenvalues (nonuniform solutions). The width of this gap depends on the scaling exponent. We associate the transition to the coherent state with the appearance of this gap.