Qualitative theory of the nonlinear behavior of coupled Josephson junctions

Abstract

We use qualitative mathematical methods for the analysis of the time dependence of the phase differences and, hence, the dynamic I‐V characteristics for two coupled Josephson point junctions. All our qualitative results are supported by numerical calculations. The case of a single point junction is reviewed and extended; it is proved that periodic trajectories in the phase plane are attractors and that they are therefore unique for a given junction and dc driving current I. Center of mass and relative coordinates are introduced for two inductively coupled junctions and the latter coordinate is shown to remain bounded. All static solutions are obtained and their stabilities analyzed. We find that the end of stability of a given solution is never a Hopf bifurcation and, hence, is marked by a finite jump to a dynamic or another static solution. The dynamic solutions are analyzed using perturbation theory for strongly coupled junctions. An approximation for the beating mode in weakly coupled junctions is presented and approximate expressions for the beat period derived. Interesting symmetries of the solution are found for a set of special values, a set which becomes dense for weak interjunction coupling.