Overview

Behavior of simple and multiple eigenvalues with a change of parameters is a problem of general interest for applied mathematics and natural sciences. This problem has many important applications in aerospace, mechanical, civil, and electrical engineering. One-parameter perturbation theory of eigenvalues for nonsymmetric matrices and differential operators was developed in [Vishik and Lyusternik (I960)], and a constructive method for determining leading terms in eigenvalue expansions was given by [Lidskii (1965)]. These works study regular types of bifurcations, when perturbation satisfies a specific nondegeneracy condition. For the analysis of some non-regular cases see [Moro et al. (1997)]. The multi-parameter bifurcation theory for eigenvalues of nonsymmetric matrices was developed in [Seyranian (1990a); Seyranian (1991a); Seyranian (1993a); Seyranian (1994a); Mailybaev and Seyranian (1999b); Mailybaev and Seyranian (2000a); Seyranian and Kirillov (2001)], where perturbations along different directions or curves in the parameter space were studied. Recent achievements of the theory of interaction of eigenvalues in multi-parameter problems are given in [Kirillov and Seyranian (2002a); Seyranian and Mailybaev (2003)].

In this chapter we present general results on bifurcation theory of multiple eigenvalues for matrices dependent on several parameters. Strong and weak interactions of eigenvalues on the complex plane are distinguished and studied. Extensions to generalized eigenvalue problem and eigenvalue problem corresponding to vibrational systems are presented. The results of this chapter represent the main tool for the multi-parameter stability analysis and are used in all parts of this book.