On the stabilization of fixed points of chaotic mappings (Q1386572)

Consider the \(m\)-dimensional mapping \((*)\) \(z_{n+1}= F(z_n,p)\) depending on the real parameter \(p\). Let \(z_* (p)\) be a fixed point of \((*)\). Suppose \(z_* (p)\) loses its stability when \(p\) crosses the critical value \(p_*\). The author proposes a state-parameter feedback to stabilize \(z_* (p)\) for \(p > p_*.\) Its basic form reads \[ z_{n+1} = F(z_n ,p)+\varepsilon (q_n - p), \qquad q_{n+1} = Q(z_n ,p) + \beta (q_n - p) + p \] whith \(Q(z_* (p),p)=0\), \(q_n \in \mathbb{R}\); \(\varepsilon \in \mathbb{R}^m\) and \(\beta \in \mathbb{R}\) are control parameters. The author studies the cases \(m=1\) and \(m=2\).