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Singularities on the boundary of the stability domain near 1:1-resonance. (English) Zbl 1193.34110

Summary: We study the linear differential equation \(\dot x = Lx \) in 1:1-resonance. That is, \(x \in \mathbb R^4\) and \(L\) is \(4\times 4\) matrix with a semi-simple double pair of imaginary eigenvalues \((i\beta , - i\beta ,i\beta , - i\beta )\). 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 \(\mathbf {gl}(4,\mathbb R)\). In this setting our aim is to determine the stability domain and the singularities of its boundary. The dimension of \(\mathbf {gl}(4,\mathbb R)\) 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 \(\mathbf {Gl}(4,\mathbb R)\). 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 \(\mathbf{gl}(4,\mathbb R)\). 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.

MSC:

34D10 Perturbations of ordinary differential equations
34A30 Linear ordinary differential equations and systems

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