Bifurcation Of Periodic Solutions Of Delay Differential Equations With Two Delays

For differential equations with one or two delays and some kinds of symmetry,the Kaplan-Yorke method is an effective tool in the study of the existence of periodic solutions with some particular periods.By applying and extending Kaplan-Yorke's method,the existence of periodic solutions for the delay differential equations and the equations with parameters and two delays is considered.Based on the qualitative analysis,it is proved that the existence of(3r₁/(3k₁+4))-periodic solution(resp.(3r'₁/(3k'₁-4))-periodic solution)for(2.1.1)(resp.(2.1.2))can be guaranteed by the existence of the periodic solution of the corresponding ordinary differential system.The conditions for the existence of(3r₁/(3k₁+4))-periodic solutions of equation(2.1.16)with parameters are also given. At last,by using the method initiated and developed by[15]and[16],that is,the generalized Hopfbifurcation theorem and the expansion of the periodic function of closed orbits, conditions for the existence of(3r₁/(3k₁+4))-periodic solution produced by the Hopf and saddle-node bifurcations are derived for equation(2.1.16).