Guiding Potentials and bounded solutions of differential equations on finite-dimensional non-compact maniforlds

The paper is devoted to some modification of the theory of guiding potentials in such a way that it becomes applicable to investigation of ordinary differential equations on non-compact on finite-dimensional non-compact smooth manifolds. Two constructions of the topological index on manifolds are described – for the maps of manifolds and for tangent and cotangent vector fields. On the basis of this modification we prove a theorem of existence of a solution that is uniformly bounded on the entire line.