TY - JOUR

T1 - Singularities on the boundary of the stability domain near 1:1 resonance

AU - Hoveijn, Igor

AU - Kirillov, Oleg

N1 - This Open Access article is freely available to read here: http://dx.doi.org/10.1016/j.jde.2009.12.004

PY - 2010/5/15

Y1 - 2010/5/15

N2 - We study the linear differential equation View the MathML source 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 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.

AB - We study the linear differential equation View the MathML source 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 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.

KW - stability domain

KW - 1:1-resonance

KW - centralizer unfolding

KW - Whitney stratification

KW - Whitney umbrella

U2 - 10.1016/j.jde.2009.12.004

DO - 10.1016/j.jde.2009.12.004

M3 - Article

VL - 248

SP - 2585

EP - 2607

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 10

ER -