Analysis for a special first order characteristic equation with delay dependent parameters

In this paper, the characteristic equation a(τ)λ + b(τ) + c(τ) e−λτ + d(τ) e−2λτ = 0 with delay τ dependent parameters is investigated. Sufficient and necessary conditions for the existence of a pair of purely imaginary roots are obtained by introducing the auxiliary equations. The crossing directions of eigenvalue curves through the imaginary axis as the delay τ increases are determined by the formulae.