A detailed analysis follows. I. Stability of invariant sets of dynamical systems in metric spaces. Stability, ordinary or asymptotic, uniform or otherwise, are all defined for closed invariant sets. Theorems stating necessary and sufficient conditions are given. They are also expressed in terms of a functional V which plays the role of the Lyapunov function in this very general context.
Application is made to distance estimates of a motion from an invariant set.
For asymptotic stability the maximal region for such stability is dealt with, and in the Euclidean space case, conditions assuring that it be the whole space are also given. Application to the ordinary differential systems. The first method of Lyapunov. This method, valid only for analytical systems, rests upon the explicit series expansions of the solutions near a singularity. It is extensively dealt with in the chapter.