Perturbation theory for a resistivity shunted Josephson element

We present a systematic perturbation theory, extendable in principle to all orders of magnitude, for the solution of the equations of motion of an ideal Josephson element shunted by a resistance and driven by a dc current source and a small time-dependent source. We present second-order results for the case in which the time dependence is that of a single sinusoid, and these results are compared with other numerical and analytical calculations. Near, but not on, the first constant voltage step where the perturbation theory appears divergent, the current-voltage characteristic is calculated by means of a nonperturbative adiabatic procedure. The impedance and responsivity agree with earlier results.