Phase locking of Josephson-junction series arrays

We report the results of a stability analysis of coherent oscillations in series arrays of Josephson junctions shunted by a common load. The analysis has found the parameter values for which the coherent, in-phase solution is stable and gives a quantitative measure of the stability. We find that arbitrarily large, dc-biased, arrays of Josephson junctions will phase lock most strongly when the capacitance parameter ${\ensuremath{\beta}}_{c}$\ensuremath{\approxeq}1 and the bias current is about the same as the critical current of the individual junctions. We use bifurcation theory to discuss the role that symmetry plays in determining the form and the stability of these oscillations. Simulations with up to 100 junctions confirm the results of the stability analysis. Arrays of nearly identical junctions are also discussed.