Smoothness loss of periodic solutions of a neutral functional differential equation: on a bifurcation of the essential spectrum

The linearized Poincare operator of a periodic solution of a neutral functional differential equation is, unlike the situation far retarded functional differential equations, no longer a compact operator. It has both a point and an essential spectrum. In the existing theory one commonly requires that the essential spectrum should be inside the unit circle and bounded away from it. However, during continuation the essential spectrum may move and approach the unit circle, causing a bifurcation that is inherently infinite-dimensional in nature since it involves an infinite number of eigenmodes. In this paper we analyse a specific system with such a bifurcation. We prove its existence and show that the smoothness of the corresponding branch of periodic solutions is lost beyond the bifurcation point.