Analytical descriptions of the geometric phases (GPs) for the total
system and subsystems are studied for a current biased Josephson
phase qubit strongly coupled to a lossy LC circuit in the dispersive
limit. It is found that, the GP and purity depend on the damping
parameter which leads to the phenomenon of GP death. Coherence
parameter delays the phenomenon of a regular sequence of deaths
and births of the GP. The asymptotic behavior of the GP and the
purity for the qubit-LC resonator state closely follow that for the
qubit state, but however, for the LC circuit these asymptotic values
are equal to zero.