Dynamics of two capacitively coupled Josephson junctions in the overdamped limit

The dynamics of two capacitively coupled Josephson junctions are investigated analytically and numerically on the basis of the RSJ-model. The attention is focussed on nearly identical junctions with zero internal capacitance, the so-called overdamped limit. This is a common assumption for high-Tc junctions. Varying one of the control currents the dynamics shows the typical phenomenon of frequency locking and the corresponding devil’s staircase for a relevant range of system parameters and upper bounds for the widths of the locking regions in terms of the control currents are derived. Correspondingly, closer inspection reveals that below a certain value of the coupling capacitance the dynamics takes place on a two-dimensional torus in the phase space. Numerical evidence is found from the calculation of Poincaré sections, and more definitely, from Lyapunov exponents. Using the concept of normal hyperbolicity, a proof of the existence of an attracting two-torus is given. Above this value the torus and the devil’s staircase partially break up and chaotic dynamics appear in between the main steps of the staircase.