On the boundaries of the parametric resonance domain

The problem of stability for a system of linear differential equations with coefficients which are periodic in time and depend on the parameters is co nsidered. The singularities of the general position arising at the boundari es of the stability and instability (parametric resonance) domains in the c ase of two and three parameters are listed. A constructive approach is prop osed which enables one, in the first approximation, to determine the stabil ity domain in the neighbourhood of a point on the boundary (regular or sing ular) from the information at this point. This approach enables one to elim inate a tedious numerical analysis of the stability region in the neighbour hood of the boundary point and can be employed to construct the boundaries of parametric resonance domains. As an example, the problem of the stabilit y of the oscillations of an articulated pipe conveying fluid with a pulsati ng velocity is considered. In the space of three parameters (the ave rage R aid velocity and the amplitude and frequency of pulsations) a singularity o f the boundary of the stability domain of the "dihedral angle" type is obta ined and the tangential cone to the stability domain is calculated. (C) 200 1 Elsevier Science Ltd. All rights reserved.