Bifurcation analysis of periodic solutions of neutral functional differential equations: A case study

This paper deals with the numerical bifurcation analysis of periodic soluti ons of a system of neutral functional differential equations (NFDEs). Compa red with retarded functional differential equations, the solution operator of a system of NFDEs does not smooth the initial data as time increases and it is no longer a compact operator. The stability of a periodic solution i s determined both by the point spectrum and by the essential spectrum of th e Poincare operator. We show that a periodic solution can change its stabil ity not only by means of a "normal" bifurcation but also when the essential spectrum crosses the unit circle. In order to monitor the essential spectr um during continuation, we derive an upper bound on its spectral radius. Th e upper bound remains valid even at paints where the radius of the essentia l spectrum is noncontinuous. This can occur when the delay and the period a re rationally dependent. Our numerical results present these new dynamical phenomena and we state a number of open questions. Although we restrict our discussion to a specific example, we strongly believe that the issues we d iscuss are representative for a general class of NFDEs.