Realization of Critical Eigenvalues for Systems of Linear Delay Differential Equations

Abstract

This thesis is a step forward in the generalization of the Realization Theorem in the paper [15] by Buono and LeBlanc. In that theorem, the two authors study the link between the number of critical eigenvalues and the number of delays in a scalar delay differential equation of the form: y&d2;t =j=1lajy t-tj,a j∈R. In this thesis, we shall consider a system of p ( p ∈ N ) scalar delay-differential equations. That system can be written as: y&d2;t =j=1lMjy t-tj,M j∈Mp R. The goal is therefore to study the links between the number of critical eigenvalues, the number of delays and the number p of equations. We study these links in three particular cases. First of all, we are interested in the case l = 1. That is y˙(t) = My( t - tau). Secondly, we consider the equation y˙ (t) = M1y( t - tau1) + M2 y(t - tau2), Mj ∈ M2R . Finally, we study the case: y&d2;t =j=1lMjy t-tj where the matrices Mj ∈ Mp R are diagonal, for 1 ≤ j ≤ l.