In this paper we develop Kaplan-Yorke's method and consider the existence of periodic solutions for some delaydifferential equations. We especially studyHopf and saddle-node bifurcations of periodic solutions with certain periods for these equations with parameters, and give conditions under which the bifurcations occur. We also give application examples and find that Hopf and saddle-node bifurcations often occur infinitelymanytimes.