Analysis for a special first order characteristic equation with delay dependent parameters (Q2468154)

The paper considers the characteristic equation \[ a(\tau) \lambda + b(\tau) + c(\tau)e^{-\lambda\tau} + d(\tau)e^{-2\lambda\tau}=0, \] which arises from a delay differential equation with delays \(\tau>0\) and \(2\tau\) and delay dependent parameters. Necessary and sufficient conditions are obtained for the existence of a pair of purely imaginary roots of this equation, i.e. for the Andronov-Hopf bifurcation in the corresponding delay equation. Additionally, the direction, in which the roots cross the imaginary axis, are indicated.