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We study synchronization in disordered arrays of Josephson junctions. In thefirst half of the paper, we consider the relation between the coupledresistively- and capacitively shunted junction (RCSJ) equations for such arraysand effective phase models of the Winfree type. We describe a multiple-timescale analysis of the RCSJ equations for a ladder array of junctions\textit{with non-negligible capacitance} in which we arrive at a second orderphase model that captures well the synchronization physics of the RCSJequations for that geometry. In the second half of the paper, motivated byrecent work on small world networks, we study the effect on synchronization ofrandom, long-range connections between pairs of junctions. We consider theeffects of such shortcuts on ladder arrays, finding that the shortcuts make iteasier for the array of junctions in the nonzero voltage state to synchronize.In 2D arrays we find that the additional shortcut junctions are only marginallyeffective at inducing synchronization of the active junctions. The differencesin the effects of shortcut junctions in 1D and 2D can be partly understood interms of an effective phase model.Comment: 31 pages, 21 figure

Synchronization in disordered Josephson junction arrays: Small-world connections and the Kuramoto model