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Bifurcations of periodic solutions of delay differential equations. (English) Zbl 1027.34081
The author extends the method of J. L. Kaplan and J. A. Yorke [J. Differ. Equations 23, 293-314 (1977; Zbl 0307.34070)] to prove the existence of periodic solutions with certain period in scalar delay differential equations of the type $$\dot x(t)= F(x(t), x(t-r), x(t-2r))$$, where $$F$$ satisfies the relation $$F(x,y,-x)=-F(-x,-y,x)$$. For $$F$$ depending on parameters, the paper gives conditions under which Hopf and saddle-node bifurcations of periodic solutions occur. Moreover, the author provides examples showing that Hopf and saddle-node bifurcations often occur infinitely many times.

##### MSC:
 34K18 Bifurcation theory of functional-differential equations 34K13 Periodic solutions to functional-differential equations
##### Keywords:
delay differential equation; periodic solution; bifurcation
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##### References:
 [1] Chen, Y., The existence of periodic solutions of the equation $$ẋ(t)=−f(x(t),x(t−τ))$$, J. math. anal. appl., 163, 227-237, (1993) · Zbl 0755.34063 [2] P. Dormayer, A.F. Ivanor, Symmetric periodic solutions of a delay differential equation, in: Dynamical Systems and Differential Equations, Vol. 1 (1998) (added volume to Discrete Continuous Dyn. Systems), Southwest Missouri State University, Springfield, pp. 220-230. · Zbl 1304.34119 [3] Ge, W., Existence of many and infinitely many periodic solutions for some types of differential delay equations, J. Beijing inst. technol., 1, 5-14, (1993) · Zbl 0807.34083 [4] Gopalsamy, K.; Li, J.; He, X., On the construction of kaplan – yorke type for some differential delay equations, Appl. anal., 59, 65-80, (1995) · Zbl 0845.34073 [5] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcation of vector fields, (1983), Springer Berlin, New York · Zbl 0515.34001 [6] Hale, J., Theory of functional differential equations, (1977), Springer Berlin, New York [7] M. Han, On the existence of symmetric periodic solutions of a differential difference equation, Differential Equations and Control Theory, Lecture Notes in Pure and Applied Math. Series, Vol. 176, Marcel Dekker, New York, 1995, pp. 73-77. [8] Jones, G.S., Periodic motions in Banach space and application to functional differential equations, Contrib. differential equations, 3, 75-106, (1964) [9] Kaplan, J.; Yorke, J., Ordinary differential equations which yield periodic solutions of differential delay equations, J. math. anal. appl., 48, 317-324, (1974) · Zbl 0293.34102 [10] Kaplan, J.; Yorke, J., On the nonlinear differential delay equation $$ẋ(t)=−f(x(t),x(t−1))$$, J. differential equations, 23, 293-314, (1977) · Zbl 0307.34070 [11] Saupe, D., Global bifurcation of periodic solutions to some autonomous differential delay equations, Appl. math. math. comput., 13, 185-211, (1983) · Zbl 0522.34067 [12] Wang, K., On the existence of nontrivial periodic solutions of differential difference equations, Chinese ann. math., 11B, 438-444, (1990) · Zbl 0729.34047 [13] Wen, L., The existence of periodic solutions for a class of differential difference equations, Chinese ann. math., 10A, 289-294, (1989)
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