Local approximations and stability: a trajectory-based approach

This paper introduces a new concept of local approximation for ordinary differential equations. Unlike linear and homogeneous approximations, which approximate the vector field near the origin, the present concept is based on approximating trajectories near the origin, which is more general. We illustrate the role of this new local approximation in stability analysis. This provides a unifying framework within which we obtain alternative proofs for and generalizations of several existing stability results