Smooth bifurcation of symmetric periodic solutions of functional differential equations (Q909092)

The delay differential equation (1) \(x'(t)=f[x(t-\alpha)]\) with odd \(f\in C({\mathbb{R}})\) is studied. A periodic solution of (1) is said to be symmetric if \(x(s+\tau)\equiv -x(s)\) for some \(\tau >0\). It is known that (1) has a continuous branch of symmetric periodic solutions, which bifurcates at \(\alpha_ 0=-f'(0)\pi^ 2\) from the trivial solution [\textit{J. L. Kaplan}, \textit{J. A. Yorke}, J. Math. Anal. Appl. 48, 317-324 (1974; Zbl 0293.34102)], and that on this (primary) branch there exists a point at which nonsymmetric periodic solution of (1) bifurcates [\textit{H. O. Walter}, Math. Z. 182, 269-289 (1983; Zbl 0488.34066)] (it is unknown if this secondary branch is smooth or not). In the paper under review the author gives conditions for the existence on the primary branch of a bifurcation point form which a smooth branch of symmetric periodic solutions bifurcates. It is also proved that \(f(x)=x/(1+x^ 2)\) satisfies these conditions.