Local-to-global coupling in chaotic maps

Abstract

We study coupled systems whose elements are chaotic maps, with the coupling ranging from "local" (with the interaction spreading over K neighbors) to "global" (mean-field-type coupling). We find that well-defined peaks emerge in the power spectrum of the mean field, indicating a subtle coherence among the elements, as the extent of coupling, i.e., K, is increased. We observe that the significant quantity here is not the ratio of the number of elements coupled K, to lattice size N, but the magnitude of K. After a critical value of K equal to K_c, the coupling takes on a "global" character, and is practically indistinguishable from mean-field interaction. Interestingly, the value of K_c≈N_c, where N_c is the critical lattice size after which the power spectra in globally coupled systems saturate. We also find that the mean-square deviation of the mean field grows linearly with coupling strength, up to K_c.