τ-D decomposition of the one dimensional differential equation with delay by fixed a>0

When we research the stability and bifurcation of one dimensional delay differential equation, the problem that is met firstly is the estimation for the roots of the characteristic equation including transcendental functions. At present, there have been many methods that can solve the above problem. But the common character of these methods is that can give the partition of the characteristic roots only in the coefficient space or delay space. In this paper, we are devoted to discuss the characteristic equation of one dimensional delay differential equation, by using τ-D decomposition. We provide a Hopf bifurcation diagram of the zero solution of the one dimensional delay differentia equation in the parameter space that is made up of the coefficient and the delay, one can determine the stability domain of the equilibrium and Hopf bifurcation curves in this parameter space according to the partition of the roots of the characteristic equation.