  I have applied multiple-scale perturbation theory to a generalized complex
$PT$-symmetric Mathieu equation in order to find the stability boundaries
between bounded and unbounded solutions. The analysis suggests that the
non-Hermitian parameter present in the equation can be used to control the
shape and curvature of these boundaries. Although this was suggested earlier by
several authors, analytic formulas for the boundary curves were not given. This
paper is a first attempt to fill this gap in the theory.
