Neimark–Sacker bifurcation for periodic delay differential equations

✍ Scribed by Gergely Röst

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 256 KB
• Volume: 60
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2004.08.043

✦ Synopsis

In this paper we study the delay differential equation

where is a real parameter, the functions a(t), f (t, ) 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.