Stability and bifurcation of a numerical discretization Mackey-Glass system

The dynamics of a kind of a discrete Mackey-Glass system obtained by Trapezium method are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold.