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Normal Forms and Bifurcations of Discrete Time Nonlinear Control Systems

The quadratic and cubic normal forms of discrete time nonlinear control systems are presented. These are the normal forms with respect to the group of state coordinate changes and invertible state feedbacks. We introduce the concept of a control bifurcation for such systems. A control bifurcation takes place at an equilibrium where there is a loss of linear stabilizability in contrast to a classical bifurcation, which typically takes place at an equilibrium where there is a loss of linear stability. We present the analogous control bifurcations to the well-known classical bifurcations; the fold, the transcritical, the flip, and the Neimark--Sacker bifurcations. When the loop is closed, a control bifurcation can lead to a classical bifurcation.