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

Hoveijn, I. ; Kirillov, O. N. (2010):
Singularities on the boundary of the stability domain near 1:1-resonance.
In: Journal of Differential Equations, 248 (10), pp. 2585-2607. Elsevier, [Article]

Item Type: Article
Erschienen: 2010
Creators: Hoveijn, I. ; Kirillov, O. N.
Title: Singularities on the boundary of the stability domain near 1:1-resonance
Language: English
Journal or Publication Title: Journal of Differential Equations
Journal volume: 248
Number: 10
Publisher: Elsevier
Divisions: 16 Department of Mechanical Engineering > Dynamics and Vibrations
16 Department of Mechanical Engineering
Date Deposited: 19 Sep 2012 14:09
Identification Number: doi:10.1016/j.jde.2009.12.004
Alternative keywords:
Alternative keywordsLanguage
Stability domain, 1:1-resonance, Centralizer unfolding, Whitney stratification, Whitney umbrellaEnglish
Alternative Abstract:
Alternative abstract Language

We study the linear differential equation x˙ = Lx in 1:1-resonance. That is, x ∈ R4 and L is 4 × 4 matrix with a semi-simple double pair of imaginary eigenvalues (iβ, −iβ, iβ, −iβ). 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 corre- spondence 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 or- bit of L under the adjoint action of Gl(4, R). Repeating a similar procedure in the versal unfolding we are able to reduce the di- mension 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 con- tained in two right conoids, one in each 3-disc. The singularities of this surface are transverse self-intersections, Whitney umbrel- las 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 con- stant on strata allows us to describe the neighborhood of L and in particular identify the stability domain.

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