An iteration method for calculating the periodic solution of time-delay systems after a Hopf bifurcation

A general description of the iteration method is presented in this paper for calculating the periodic solution resulted from a Hopf bifurcation of time-delay systems including the degenerated cases (the delays disappear): ordinary differential equations. Two algorithms are developed for scalar systems and for general systems, respectively. For scalar systems, the iteration method is straightforward, and for general systems, the method needs to solve two eigenvalue problems before the construction of the straightforward iteration scheme. As shown in the four illustrative examples, the iteration method works effectively. It involves easy computation only, the first iteration is usually enough for achieving the accurate bifurcation direction and an accurate estimation of the bifurcated periodic solution.