Areas of attraction for nonautonomous differential equations on finite time intervals

In this article, new concepts of (exponential) attractivity for nonautonomous differential equations on a finite time interval are introduced. Due to nonuniqueness of finite-time objects, areas of attraction are studied rather than solutions which are attractive. A sufficient and necessary condition for the existence of such areas is presented, which makes use of a time-varying Riemannian metric. Several examples illustrate the theoretical results and definitions.