Bifurcation analysis of delayed logistic equation

Using the Liapunov-Schmidt reduction, we investigate the Hopf bifurcation of the well-known delayed logistic equation. Near the Hopf bifurcation point, we obtain the periodic solutions branch bifurcated from the trivial solution. The approximate analytic expressions of the periodic solutions are given to compare with the numerical results, which are computed by the collocation method based on piesewise Hermite polynomials. The fact that the approximate analytic periodic solutions nearly coincides with the numerical results shows the effectiveness of our analysis.