Controllability near a homoclinic bifurcation

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