On stability-preserving mappings of dynamical systems. I: Theory
In this paper, a general dynamical system \(S\) is defined as a family of maps \(p : \mathbb{R}^+ \times A \times \mathbb{R}^+ \to X\), where \(X\) is a metric space, \(A \subseteq X\), \(p(t,a,t_0)\) is defined for all \(t \geq t_0\) and \(p(t_0, a, t_0) = a\) \((\mathbb{R}^+\) can be replaced by \(\mathbb{N}\), so that continuous time and discrete time systems are treated simultaneously). For a set \(M \subseteq X\) which is invariant with respect to all \(p \in S\), several notions of stability are given. These include generalizations of classical notions like uniform stability, asymptotic stability, exponential stability, boundedness of solution and so on. Given two dynamical systems \(S_1\) and \(S_2\) on metric spaces \(X_1\) and \(X_2\) respectively, assume that there exists \(V : X_1 \times \mathbb{R}^+ \to X_2\) which maps \(S_1\) on \(S_2\). Assume that \(M_1 \subseteq X_1\), \(M_2 \subseteq X_2\) are invariant with respect to \(S_1\), \(S_2\) respectively, and assume further that \(V\) maps \(M_1\) on \(M_2\). The authors prove a sufficient condition for \(V\) being stability preserving, which to say that \(S_1\) possesses one of the aforementioned stability properties if and only if \(S_2\) possesses the same property.
scientific article; zbMATH DE number 919491