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A mystery surrounds the stability properties of the splay-phase periodicsolutions to a series array of N Josephson junction oscillators. Contrary towhat one would expect from dynamical systems theory, the splay state appearsto be neutrally stable for a wide range of system parameters. It has beenexplained why the splay state must be neutrally stable when theStewart-McCumber parameter beta is zero. In this paper we complete theexplanation of the apparent neutral stability; we show that the splay stateis typically hyperbolic -- either asymptotically stable or unstable -- whenbeta > 0. We conclude that there is only a single unit Floquet multiplier,based on accurate and systematic computations of the Floquet multipliers forbeta ranging from 0 to 10. However, N-2 multipliers are extremely close to 1for beta larger than about 1. In addition, two more Floquet multipliersapproach 1 as beta becomes large. We visualize the global dynamicsresponsible for these nearly degenerate multipliers, and then estimate themaccurately by a multiple time-scale analysis. For N=4 junctions the analysisalso predicts that the system converges toward either the in-phase state, thesplay state, or two clusters of two oscillators, depending on the parameters.

Stability of Periodic Solutions in Series Arrays of Josephson Junctions with Internal Capacitance