Controlling and anti-controlling Hopf bifurcations in discrete maps using polynomial functions

In this paper, we consider controlling and anti-controlling Hopf bifurcations in discrete maps using feedback controller of polynomial functions. It is shown that such a polynomial feedback controller is easy to be implemented, which not only preserves the system's equilibrium solutions, but also keeps the dimension of the system unchanged. Examples are used to show, with this type of controller, that one may effectively control Hopf bifurcation (e.g., delaying the onset of an existing Hopf bifurcation) and anti-control Hopf bifurcation (e.g., generating a Hopf bifurcation as desired) in two-dimensional and higher dimensional discrete maps. Results can be extended to control other types of bifurcations, such as Hopf-zero and double-Hopf, etc. bifurcations in discrete maps.