Prolongation and stability concepts in dynamical system (Q2714468)

It is known that in the study of topological properties of ordinary differential equations the stability theory of compact invariant sets (which may be regarded as generalization of critical points and limit cycles) plays a central role. In this note the authors show that if \(M\) is a compact subset of a Hausdorff space, then \(D^+(M)\) that is the first positive prolongation is closed. The authors show that, if \(M\) is stable, then \(D^+(M)=M\). Moreover they give a characterization of stability by the first negative prolongation of its complement.