Josephson-junction circuit analysis via integral manifolds

Using a second-order circuit model the complex dynamical behavior of a typical Josephson-junction circuit is rigorously analyzed using integral manifolds. The key idea is to prove that under certain small-parameter assumptions, the nonautonomous circuit has a stable integral manifold. Moreover, this manifold is doubly periodic so that steady-state behavior of the Josephson-junction circuit reduces to the analysis of its dynamics on a torus. Well-known experimental phenomena, such as the existence of hysteresis in the dc Josephson circuit and voltage steps in the ac Josephson circuit, are rigorously derived and explained.