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We present simulations of two-dimensional Josephson-junction arrays to study giant Shapiro steps in these arrays. The amplitude and frequency dependence of the step widths is found to be more complex than in single junctions. The fractional step widths are found to decrease more rapidly with increasing frequency or rf current than conventional steps in single junctions. The washboard model of single junctions is extended to arrays to explain these differences between arrays and single junctions.

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The stability of the periodic solution of the two-cell ladder array has been numerically investigated in order to explore intrinsic phase-locking mechanisms relevant to arrays and stacked junction oscillators. In zero magnetic field the periodic in-phase solution of the system is neutrally stable. However, this solution is stable over a finite voltage range when an applied control current exceeds a critical value. The dependence upon system parameters of the boundaries of the stable range and the critical control current is investigated. Finally, the influence of the control current on the microwave power in a typical range of stability is calculated.

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