Singularities on the boundary of the stability domain near <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns
Singularities on the boundary of the stability domain near <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns
复制标题
DOI：
10.1016/j.jde.2009.12.004
复制
发表时间：
2010-05
影响因子：
2.4
通讯作者：
I. Hoveijn;Oleg N. Kirillov
中科院分区：
文献类型：
--
作者：
I. Hoveijn;Oleg N. Kirillov
我们研究了1:1共振中的线性微分方程x˙=Lx。即，x∈r4， L是具有半简单双虚特征值对（iβ, - iβ,iβ, - iβ）的4×4矩阵。我们希望找到这个线性系统的所有摄动，使摄动系统是稳定的。由于线性微分方程与线性映射是一一对应的，我们把这个问题转化为gl（4，R）。在这种情况下，我们的目的是确定稳定域及其边界的奇异性。gl（4，R）的维数为16，因此我们首先尽可能地减少维数。这里我们使用L的一般展开，即L的轨道在Gl（4，R）的伴随作用下的横切面。在一般展开中重复类似的过程，我们能够将维数降至4。这个四维空间中的一个3球包含了gl（4，R）中L的邻域的所有信息。将3球视为沿其共同边界光滑粘接的两个3盘，我们发现稳定域的边界包含在两个右圆锥体中，每个3盘中有一个。这个曲面的奇点是横向自交，惠特尼伞和自交的交集，其中曲面有一个自切线。3球的惠特尼分层使得相应矩阵的特征值构型在地层上是恒定的，这使我们能够描述L的邻域，特别是识别稳定域。
We study the linear differential equation x˙=Lx in 1:1-resonance. That is, x∈R4and 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.