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

I. Hoveijn*, O. N. Kirillov

*Corresponding author for this work

    Research output: Contribution to journalArticleAcademicpeer-review

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    We study the linear differential equation (x) over dot = Lx in 1:1-resonance. That is, x is an element of R(4) and L is 4 x 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 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 GI(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 colloids, 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. (C) 2009 Elsevier Inc. All rights reserved.

    Original languageEnglish
    Pages (from-to)2585-2607
    Number of pages23
    JournalJournal of Differential Equations
    Issue number10
    Publication statusPublished - 15-May-2010


    • Stability domain
    • 1:1-resonance
    • Centralizer unfolding
    • Whitney stratification
    • Whitney umbrella
    • ORBITS

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