  A singular perturbation problem from the artificial compressible system to
the incompressible system is considered for a doubly diffusive convection when
a Hopf bifurcation from the motionless state occurs in the incompressible
system. It is proved that the Hopf bifurcation also occurs in the artificial
compressible system for small singular perturbation parameter, called the
artificial Mach number. The time periodic solution branch of the artificial
compressible system is shown to converge to the corresponding bifurcating
branch of the incompressible system in the singular limit of vanishing
artificial Mach number.
