  Controllability properties are studied for control-affine systems depending
on a parameter and with constrained control values. The uncontrolled systems in
dimension two and three are subject to a homoclinic bifurcation. This generates
two families of control sets depending on a parameter in the involved vector
fields and the size of the control range. A new parameter given by a split
function for the homoclinic bifurcation determines the behavior of these
control sets. It is also shown that there are parameter regions where the
uncontrolled equation has no periodic orbits, while the controlled systems have
periodic solutions arbitrarily close to the homoclinic orbit

Optimization and Control