  We study the linear differential equation x' = Lx in 1:1 resonance. That is,
x in R^4 and L is a 4 by 4 matrix with a semi-simple double pair of imaginary
eigenvalues (ib,-ib,ib,-ib). We wish to find all perturbations of this linear
system such that the perturbed system is stable. Since linear differential
equations are in one to one correspondence with linear maps we translate this
problem to gl(4,R). In this setting our aim is to determine the stability
domain and the singularities of its boundary. The dimension of gl(4,R) is 16,
therefore we first reduce the dimension as far as possible. Here we use a
versal unfolding of L ie a transverse section of the orbit of L under the
adjoint action of Gl(4,R). Repeating a similar procedure in the versal
unfolding we are able to reduce the dimension to 4. A 3-sphere in this
4-dimensional space contains all information about the neighborhood of L in
gl(4,R). In this 3-sphere the boundary of the stability domain is a surface
with singularities: transverse self-intersections, Whitney umbrellas and an
intersection of self-intersections where the surface has a self-tangency.
