  Sufficient and necessary conditions are established for controllability of
affine control systems, motivated by these with a drift all of whose solutions
are periodic. In contrast with previously known results, these conditions
encompass the case of a control set whose convex hull contains the origin but
is not a neighborhood of the origin. The condition is expressed by means of
pushforwards along the flow of the drift, rather than in terms of Lie brackets.
It turns out that this also amounts to local controllability of a time-varying
linear approximation with constrained controls. Global and local results are
given, as well as a few illustrative examples.
