  We use numerical continuation and bifurcation techniques in a boundary value
setting to follow Lyapunov families of periodic orbits. These arise from the
polygonal system of $n$ bodies in a rotating frame of reference. When the
frequency of a Lyapunov orbit and the frequency of the rotating frame have a
rational relationship then the orbit is also periodic in the inertial frame. We
prove that a dense set of Lyapunov orbits, with frequencies satisfying a
diophantine equation, correspond to choreographies. We present a sample of the
many choreographies that we have determined numerically along the Lyapunov
families and along bifurcating families, namely for the cases $n=4,~6,~7,~8$,
and $9$. We also present numerical results for the case where there is a
central body that affects the choreography, but that does not participate in
it. Animations of the families and the choreographies can be seen at the link:
http://mym.iimas.unam.mx/renato/choreographies/index.html
