Neimark-Sacker bifurcation for periodic delay differential equations

In this paper we study the delay differential equationx˙(t)=γ(a(t)x(t)+f(t,x(t-1))),where γ is a real parameter, the functions a(t), f(t,ξ) are C4-smooth and periodic in the variable t with period 1. Varying the parameter, eigenvalues of the monodromy operator (the derivative of the time-one map at the equilibrium 0) cross the unit circle and bifurcation of an invariant curve occurs. To detect the critical parameter-values, we use Floquet theory. We give an explicit formula to compute the coefficient that determines the direction of the bifurcation. We extend the center manifold projection method to our infinite-dimensional Banach space using spectral projection represented by a Riesz-Dunford integral. The Neimark-Sacker Bifurcation Theorem implies the appearance of an invariant torus in the space C×S1. We apply our results to an equation used in neural network theory.