Stability for dynamical systems with first integrals: A topological criterion (Q1205733)

Let \(\Phi\) denote a continuous flow on an open set of \(\mathbb{R}^ n\) containing the origin \({\mathcal O}\). Assume that \({\mathcal O}\) is an equilibrium point for the flow. Further, assume that \(\Phi\) possesses a set of \(k\) continuous first integrals, \(G(x)=(G_ 1(x),\dots,G_ k(x))\), and denote by \(\Phi^ h\) the restriction of \(\Phi\) to the level set \(G(x)=h\). The authors prove that if \({\mathcal O}\) is an asymptotically stable equilibrium of \(\Phi^ 0\) (with respect to perturbations restricted to \(\Phi^ 0\)) then it is a stable equilibrium of the flow \(\Phi\).