Mathematics – Dynamical Systems
Scientific paper
2009-11-06
Journal of Differential Equations, 248, 2585--2607 (2010)
Mathematics
Dynamical Systems
22 pages, 8 figures
Scientific paper
10.1016/j.jde.2009.12.004
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.
Hoveijn Igor
Kirillov Oleg N.
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