Sufficient conditions for the asymptotic stability of invariant manifolds
Consider the nonlinear system \(\dot x = f(t,x)\), \((t,x) \in \Omega \subset R^ 1 \times R^ n\), and let \(\varphi (t,t_ 0, x_ 0)\) be a solution of this system passing through the point \((t_ 0,x_ 0) \in \Omega\). An invariant manifold \(S\) of the nonlinear system is called strongly asymptotically stable if it is asymptotically stable and there exists a neighbourhood \(V\) of \(S\), such that \(\forall (t_ 0, x_ 0) \in V\) and \(\forall t \geq t_ 0\) the solution \(\varphi (t,t_ 0, x_ 0)\) is in the \(\delta_ 0\)-neighbourhood \(V_ 0\) and there exists a solution \(\varphi (t,t_ 0, p_ \infty)\) lying on \(S\) for which \(\| \varphi (t,t_ 0, x_ 0) - \varphi (t,t_ 0, p_ \infty) \| \to 0\) as \(t \to \infty\). In the present paper, there are given some sufficient conditions for the strong asymptotic stability of \(S\). A special case of these conditions is the well-known Poincaré criterion for the asymptotic orbital stability of a limit cycle.
scientific article; zbMATH DE number 716363