The controlled center dynamics of discrete time control bifurcations

In this paper, we introduce the Controlled center dynamics for nonlinear discrete time systems with uncontrollable linearization. This is a reduced order control system whose dimension is the number of uncontrollable modes and whose stabilizability properties determine the stabilizability properties of the full order system. After reducing the order of the system, the synthesis of a stabilizing controller is performed based on the reduced order control system. By changing the feedback, the stability properties of the controlled center dynamics will change, and thus the stability properties of the full order system will change too. Thus, choosing a feedback that stabilizes the controlled center dynamics allows stabilizing the full order system. This approach is a reduction technique for some classes of controlled differential equations.