A new unified concept of stability

In the useful and rich field of stability theory for nonlinear systems, there have been many refinements, extensions and generalizations [3, 4]. Basically, stability concerns with comparing phase-space positions of solutions of perturbed and unperturbed equations, with classical Lyapunov stability being too stringent a requirement and orbital stability being too loose a demand. We define a new concept of stability that can unify theses two extreme cases (and possibly many other appropriate notions between these two) in terms of suitable topologies, following the idea of J.L. Massera [6]. Also, a further unification is achieved by using two measures [5]. In this unified frame work, we give sufficient conditions for these concepts to hold, via Lyapunov functions.