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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 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:
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations