Bifurcations on Control-Transverse Dynamics

To every submanifold transverse to the control distribution of a given control system is associated in a canonical way a dynamical behavior. By varying this transverse manifold, we can effect bifurcations in these dynamics. In this paper we explain how this is done and draw the analogies with the more familiar aspects of bifurcation theory. The second crucial point is that the manifold can be made invariant for a control flow and the dynamics in the directions transverse to it can be assigned arbitrarily — in particular, the manifold can be made asymptotically stable.