  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.
