Neimark–Sacker bifurcation for periodic delay differential equations

In this paper we study the delay differential equation ẋ(t)= (a(t)x(t)+ f (t, x(t − 1))), where is a real parameter, the functions a(t), f (t, ) 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. 2004 Elsevier Ltd. All rights reserved. MSC: primary 34K18; 34K05; secondary 47A10; 37M20