Study of main scenarios of bifurcation for functional differential time-delay equations

In the paper, we study the main bifurcation scenarios for functional differential time delay equations with periodic right side and for nonlinear autonomous equations with aftereffect. The main tool is the operator method for studying multi-parameter bifurcation leading us to new sufficient bifurcation conditions and allowing us to obtain the approximate formulae for appearing solutions. As applications, we consider the problems on bifurcation points for the modifications of Duffing equation and Hutchinson–Wright equation.