On the (in)stability of quasi-static paths of smooth systems: definitions and sufficient conditions

A concept of stability of quasi-static paths is discussed in this paper that takes into account the existence of fast (dynamic) and slow (quasi-static) time scales in the evolution of many mechanical systems. The proposed concept is essentially a continuity property with respect to the smallness of the initial perturbations (as in Lyapunov stability) and the smallness of the quasi-static loading rate (that plays the role of the small parameter in singular perturbation problems). A related concept of attractiveness is also proposed. Several examples illustrate the relevance of the definitions. Sufficient conditions for attractiveness or for instability of quasi-static paths of smooth systems are proved.