Overview

Finding the stability and instability domains in the parameter space is the main problem for the stability theory of periodic systems. Usually this problem is solved by constructing the stability boundary and specifying the type of instability for each part of the boundary. We know that the stability boundary in the parameter space consists of smooth surfaces, but may have singularities. The simplest singularities are angles appearing in two-dimensional parameter space, but more complicated singularities can occur in multi-parameter spaces. These singularities reflect physical properties of the underlying system, and their study requires special treatment based on the bifurcation theory approach. Naturally, we are mostly interested in analyzing generic (typical) singularities of the stability boundary.

In this chapter, following [Mailybaev and Seyranian (2000a); Mailybaev and Seyranian (2000b)] we describe the stability boundary for a general linear system of ordinary differential equations with periodic coefficients dependent on real parameters. Regular part of the stability boundary corresponding to parametric and combination resonances is described, and its first and second order approximations are derived using derivatives of simple multipliers. Classification of generic singularities of the stability boundary for two- and three-parameter periodic systems is given, and the formulae for first order approximations of the stability domain near the singularities are derived. These formulae have a constructive form and require the information only at the singularity point: values of multipliers, eigenvectors, matriciants, and derivatives of the system matrix with respect to parameters. The suggested approach is useful for numerical construction and analysis of...