Delay feedback control for interaction of hopf and period doubling bifurcations in discrete-time systems

Interaction of Hopf and period doubling bifurcations, as a so-called codimension two singularity, may give rise to rich bifurcation outcomes depending on the two-parameter unfolding at the bifurcation point. In this paper, we develop a nonlinear delay control for interaction of Hopf and period doubling bifurcations at a desired parameter location with preferred bifurcation properties. For its linear control term, the gains are determined by the bifurcation critical conditions and are used to create this type of bifurcations at a desired parameter location. For its nonlinear control term, the gains are derived by the center manifold theorem and theory of normal form. They determine the type and stability of bifurcation solutions. The proposed control method is applicable to any uncontrolled system which can be stable, unstable or even chaotic. Numerical experiments verify the feasibility of the control methodology and clearly show easy manipulation of all kinds of possible solutions of the created bifurcation.