Overview

A wide range of practical problems in mechanics and physics requires stability analysis of a linear system of ordinary differential equations, which appear as a result of linearization of equations of motion near a stationary solution or steady motion. Any physical system depends on parameters, and values of the parameters at which the system is stable form the stability domain in the parameter space. It is clear that construction of the stability domain is closely related to finding its boundary.

Analysis of the stability domain and its boundary is a problem of great practical importance. A number of examples reveal complexity of the stability boundary, which consists of smooth parts and can have different singularities. The singularities are related to bifurcations of eigenvalues of the system operator. They reflect specific physical properties of the system and may lead to numerical difficulties of the analysis. Classification of singularities of the stability boundary for two- and three-parameter systems was done in [Arnold (1972); Arnold (1983a)], and the extension to a more general case was given in [Levantovskii (1980a); Levantovskii (1982)]. Quantitative methods of stability analysis near regular and singular points of the stability boundary were developed in [Seyranian (1982); Pedersen and Seyranian (1983); Burke and Overton (1992); Mailybaev (1998); Mailybaev and Seyranian (1998b); Mailybaev and Seyranian (1999b); Mailybaev (1999)].

This chapter is devoted to stability analysis of a general linear system of ordinary differential equations, whose coefficients are smooth functions of parameters. First, we introduce the concept of general position. This...