Stability and chaos of a driven nano-electromechanical Josephson junction

We consider the motion of and Josephson current through a mechanically oscillating superconducting island asymmetrically embedded in a Josephson junction. The electromechanical coupling is provided by distance dependent tunneling rates between the electrodes and the island. The system asymmetry, resulting from the geometrical configuration, leads, for weak coupling, to an equation of the mechanical motion that reduces to the well-known Duffing equation. At zero bias voltage the island motion is determined by the homogenous Duffing equation that opens up two separate regions of solutions depending on the superconducting phases. The island either moves under influence of an anharmonic single well potential, or is governed by a double well potential that allows for off-center oscillations. Under applied bias voltage the island equation of motion turns into a modified Duffing equation, with time dependent coefficients, that demonstrate both quasi periodic and chaotic behavior.