Neimark–Sacker bifurcation for periodic delay differential equations

Abstract

In this paper we study the delay differential equation x ˙ ( 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 C 4 -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 × S 1 . We apply our results to an equation used in neural network theory.