Neimark-Sacker bifurcation for periodic delay differential equations (Q1764856)

The paper considers nonautonomous delay differential equations of the form \[ x'(t)=\gamma(a(t)x(t)+f(t,x(t-1))), \] where \(a\) and \(f\) are \(1-\)periodic with respect to \(t\). The aim is to study Neimark-Sacker bifurcations. The method employed is based on characteristic equations and Floquet exponents. The methods enable the computation of critical parameter values, where the bifurcation arises, and the coefficient that determines the nature of the bifurcation. The authors are able to give two new theorems, one on the direction of the invariant curve and the other on the spectral projection operator. The results are illustrated in the final section of the paper, where an example drawn from neural networks is the subject of study.