Periodic solutions of a planar delay equation

Abstract

SynopsisWe study the planar delay differential equation x′(t) = −x(t) + αF(x(t − 1)), for α > 0. An existence theorem for nonconstant periodic solutions is achieved for a certain class of maps F, for α > some α0. Besides a condition of nondegeneracy at x = 0, we assume F is bounded and satisfies a kind of planar negative feedback condition. The nonconstant periodic solutions are associated with nontrivial fixed points of a certain operator defined by the flow in the plase space C([−l, 0], R2). In our approach, the existence of such fixed points depends on the ejectivity of O ϵ C([−1, 0], R2) with respect to that operator. Relaxing the boundedness condition on F, we show the existence of a sequence of values of α, α0 < α1 <…, where a Hopf bifurcation occurs.