I. Introduction and Motivations
A well established technique for the study of stability and robustness of nonlinear systems, which are described by a set of differential equations globally defined in Euclidean space, is the Input-to-State Stability approach, (see [17] and references therein). The classical definition potentially allows to formulate and characterize stability properties with respect to arbitrary compact invariant sets (and not simply equilibria). The implicit requirement that these sets should be simultaneously Lyapunov stable and globally attractive, however, makes the basic theory not applicable for a global analysis of many dynamical behaviours of interest, such as multistability or periodic oscillations, just to name a few, and only local analysis remains possible [6]. In fact, it is well-known that such systems, when defined in Euclidean space, normally admit invariant sets (such as additional equilibria) that fail to be Lyapunov stable.