  For control-affine systems on non-compact manifolds, the notion of strong
chain control sets is introduced and related to the strong chain transitivity
of the associated control flows. Affine control systems on R^n are embedded
into bilinear control systems in an extended state space and it is shown that
they are topologically conjugate to the induced system on the northern
hemisphere of the Poincar\'e sphere. This preserves strong chain control sets.
Further chain controllability properties on spheres are analyzed
