On gradient dynamical system on semi-Riemannian manifolds

Abstract

We consider gradient dynamical systems on a semi-Riemannian manifold of arbitrary index. The main point of the paper is the introduction of the concepts causality subsets, causality function and sector stability. As a main application we provide conditions assuring, that the nonwandering points are precisely the singular points of the gradient field. Furthermore we show, that every nonconstant recurrent orbit for the gradient field must intersect one of the causality subsets and that the stable and unstable manifolds belonging to a hyperbolic singular point for the gradient field are orthogonal.