The stabilization of unstable cycles of chaotic mappings (Q5934153)

The author considers a multidimensional mapping \[ z_{n+1}= F(z_n,p),\;z\in\mathbb R^n,\;p\in\mathbb R \tag{1} \] that is sufficiently smooth and depends on \(p\). The goal of the author is to localize and stabilize an unstable \(k\)-tuple cycle of (1) in the domain \(p>p_c\) \((p_c\) is a critical value) by small perturbations of the system parameter. For this he uses the method for localizing and stabilizing fixed points of a chaotic mapping. For \(m=1\) and \(m=2\) specific formulation of the results is given and the method is applied to the logistic map and the Hénon transformation.