Neimark–Sacker bifurcation for periodic delay differential equations

Röst, Gergely (2005) Neimark–Sacker bifurcation for periodic delay differential equations. Nonlinear Analysis: Theory, Methods and Applications, 60 (6). pp. 1025-1044. ISSN 0362-546X

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Abstract

In this paper we study the delay differential equation \dot{x}(t) = gamma(a(t)x(t) + f(t, x (t - 1))), where gamma is a real parameter, the functions a(t), f(t,xi) 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 CxS(1). We apply our results to an equation used in neural network theory.

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