Dynamics and chaos of a current driven two-dimensional Josephson junctions array under magnetic field

We derive the equations of motion for a two-dimensional capacitive Josephson junctions array in the presence of both a DC current and a magnetic field of the quantum flux Φ0 . The ground state symmetry of an N×N array is assumed to hold for all currents, then by using the resistively and capacitively shunted junction equations, a model system of four-coupled non-linear second-order differential equations is derived. The system has the form , where U is a four-dimensional potential and βc is the Stewart–McCumber parameter. The dynamics can be viewed as the motion of a massive particle sliding under the action of the potential in a four-dimensional configuration space with a friction proportional to its speed. There are three distinct branches: one below the critical current Ic where the static zero voltage solution is stable; the second branch which originates from the static solution through a Hopf bifurcation and where a total voltage develops along the direction of the applied current and across the array (instantaneous Hall voltage), (the latter means vortices moving perpendicular to the current and constitutes a flux-flow like regime); and a third branch above the synchronization current Is, where the motion of the junctions synchronizes and the motion of the vortices ceases with zero Hall voltage. For a wide range of βc, the second branch shows chaotic dynamics of extremely rich complexity. A pervasive feature is the presence of antimonotonicity, i.e., reversals of period doubling cascades.