Singularities on the boundary of the stability domain near 1:1 resonance

TL;DR

This paper analyzes the stability boundary of linear systems near 1:1 resonance, identifying singularities such as Whitney umbrellas and self-intersections through geometric and algebraic reduction techniques.

Contribution

It introduces a method to reduce the high-dimensional stability problem to a 3-sphere, revealing the structure and singularities of the stability boundary near resonance.

Findings

Identified singularities on the stability boundary, including Whitney umbrellas.

Reduced the problem dimension from 16 to 4 using versal unfolding.

Mapped the stability boundary's complex geometry in a 3-sphere.

Abstract

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.