On singularities of a boundary of the stability domain

This paper deals with the study of generic singularities of a boundary of t he stability domain in a parameter space for systems governed by autonomous linear differential equations (y) over dot = Ay or x((m)) + a(1)x((m-1)) ...+a(m)x = 0. It is assumed that elements of the matrix A and coefficients of the differential equation of mth order smoothly depend on one, two, or three real parameters. A constructive approach allowing the geometry of sin gularities (orientation in space, magnitudes of angles, etc.) to be determi ned with the use of tangent cones to the stability domain is suggested. The approach allows the geometry of singularities to be described using only f irst derivatives of the coefficients a(i) of the differential equation or f irst derivatives of the elements of the matrix A with respect to problem pa rameters with its eigenvectors and associated vectors calculated at the sin gular points of the boundary. Two methods of study of singularities are sug gested. It is shown that they are constructive and can be applied to invest igate more complicated singularities for multiparameter families of matrice s or polynomials. Two physical examples are presented and discussed in deta il.