On control of chaos: Higher periodic orbits (Q1905819)

For a given dynamical system \(x_{k + 1} = f(x_k) + Bu_k\) the problem is to find the linear state feedback \(u_k = K^T (x_k - x^*)\) such that the system is stable in the fixed point \(x^*\). There is no major difference between stabilizing the system around the unstable \(m\)-periodic point and stabilizing it around an unstable fixed point. It suffices to consider the new system \(x_{k + 1} = f^m (x_k) + Bv_k\); for the fixed point, \(m = 1\). The examples are presented for the Hénon map, including the period-13 orbit and the Ikeda map.