  A singularly perturbed system for doubly diffusive convection equations,
called the artificial compressible system, is considered on a two-dimensional
infinite layer for a parameters range where the Hopf bifurcation occurs in the
corresponding incompressible system. The spectrum of the linearized operator in
a time periodic function space is investigated in detail near the bifurcation
point when the singular perturbation parameter is small. The results of this
paper are the basis of the study of the nonlinear Hopf bifurcation problem and
the singular limit of the time periodic bifurcating solutions.
