A Geometric Approach to Global-Stability Problems

A new criterion for the global stability of equilibria is derived for nonlinear autonomous ordinary differential equations in any finite dimension based on recent developments in higher-dimensional generalizations of the criteria of Bendixson and Dulac for planar systems and on a local version of the $C^1 $ closing lemma of Pugh. The classical result of Lyapunov is obtained as a special case.