Bifurcation analysis in a scalar delay differential equation

Abstract

The dynamics of a scalar delay differential equation, which includes Mackey–Glass equations, are investigated. We prove that a sequence of Hopf bifurcations occurs at the equilibrium as the delay increases. An explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions is derived, using the theory of normal form and centre manifold. The global existence of multiple periodic solutions is established using a global Hopf bifurcation result given by Wu (1998 Trans. Am. Math. Soc. 350 4799) and a Bendixson criterion for higher dimensional ordinary differential equations given by Li and Muldowney (1993 J. Diff. Eqns 106 27).