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

The linearized Poincare operator of a periodic solution of a neutral functi onal 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 tha t the essential spectrum should be inside the unit circle and bounded away from it. However, during continuation the essential spectrum may move and a pproach the unit circle, causing a bifurcation that is inherently infinite- dimensional in nature since it involves an infinite number of eigenmodes. I n 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.