A New Convex Controller for Stabilizing of Two Symmetrical Logistic Maps

Abstract

In this paper, we focused on the coupled logistic map, which is a system of two symmetrically identical single logistic maps. The complicated dynamics of the generated system, which has a wide range of dynamical behavior, grant it much attention for many applications. First, the existence and uniqueness of this system are investigated, where the analytical technique is utilized to find the parametric condition for the local asymptotic stability of its fixed points. It can be obviously seen that it undergoes Niemark-Sacker and Hopf bifurcation in a small neighborhood of the unique positive fixed point and invariant circle. This has been proven based on bifurcation theory and the center manifold theorem. The behavior of this system is stabilized using two conventional controlling methods, the OGY and the pole-placement. As well as, a new control method is proposed to make such stabilization more easier. The numerical simulations are utilized to demonstrate the analytic results.