Limit-cycle-like control for 2-dimensional discrete-time nonlinear control systems and its application to the Hénon map

In this paper, a control method that generates a desired limit-cycle-like behavior for a 2-dimensional discrete-time nonlinear control system is discussed. First, we define some notations and state the problem formulation. Next, a necessary and sufficient condition of existence of a control input that realizes a desired limit-cycle-like behavior is shown. We then derive a control algorithm to solve the problem on generating limit-cycle-like behaviors, and a modification of the algorithm is also shown. Finally, we apply the two types of algorithms to a chaotic system, the Hénon map, in order to indicate the availability of the proposed method. In addition, by using the control method, we also consider a stabilization problem for the Hénon map.

► Some concepts on discrete-time limit cycle are formulated.
► A necessary and sufficient condition on the existence of a control input is obtained.
► Algorithms to generate a desired limit-cycle-like behavior are shown.
► An Application to the Henon map shows the effectiveness of our new method.