Determining the Stability Domain of Perturbed Four-Dimensional Systems in 1:1 Resonance

For a parametric family of four-dimensional linear dynamical systems determined by a matrix A(m) with A(0) in 1 : 1 semi-simple resonance, we have established that the central singularity on the stability boundary has codimension 8, ie the centralizer unfolding of the family needs 8 parameters. By recognizing equivalence classes in the centralizer unfolding we reduced the codimension to 5 and finally by using the homogeneity properties to 3. This allowed us to find explicitly the boundary of the stability domain and list all its singularities including six self-intersections and four ’Whitney umbrellas’. We have proposed an algorithm of approximation of the stability boundary near singularities and applied the results to the study of enhancement of the modulation instability with dissipation as well as to the study of stability of a nonconservative system of rotor dynamics.