Bifurcation Analysis of Periodic Solutions of Neural Functional Differential Equations: A Case Study

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Abstract
This paper deals with the numerical bifurction analysis of periodic solutions of a system of neutral functional differential equations (NFDEs). Compared 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 is determined both by the point spectrum and by the essential spectrum of the Poincaré operator. We show that a periodic solution can change its stability not only by means of a "normal" bifurcation but also when the essential spectrum crosses the unit circle. In order to monitor the essential spectrum during continuation, we derive an upper bound on its spectral radius. The upper bound remains valid even at points where the radius of the essetial spectrum is noncontinuous. This can occur when the delay and the period are 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 discuss are representative for a general class of NFDEs.