Singularities on the boundary of the stability domain near [formula omitted]-resonance

Abstract

We study the linear differential equation x ˙ = L x in 1 : 1 -resonance. That is, x ∈ R 4 and L is 4 × 4 matrix with a semi-simple double pair of imaginary eigenvalues ( i β , − i β , i β , − i β ) . 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, i.e. 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 ) . Considering the 3-sphere as two 3-discs glued smoothly along their common boundary we find that the boundary of the stability domain is contained in two right conoids, one in each 3-disc. The singularities of this surface are transverse self-intersections, Whitney umbrellas and an intersection of self-intersections where the surface has a self-tangency. A Whitney stratification of the 3-sphere such that the eigenvalue configurations of corresponding matrices are constant on strata allows us to describe the neighborhood of L and in particular identify the stability domain.