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We use numerical continuation and bifurcation techniques in a boundary valuesetting to follow Lyapunov families of periodic orbits. These arise from thepolygonal system of $n$ bodies in a rotating frame of reference. When thefrequency of a Lyapunov orbit and the frequency of the rotating frame have arational relationship then the orbit is also periodic in the inertial frame. Weprove that a dense set of Lyapunov orbits, with frequencies satisfying adiophantine equation, correspond to choreographies. We present a sample of themany choreographies that we have determined numerically along the Lyapunovfamilies 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 acentral body that affects the choreography, but that does not participate init. Animations of the families and the choreographies can be seen at the link:http://mym.iimas.unam.mx/renato/choreographies/index.html

Symmetries and choreographies in families bifurcating from the polygonal relative equilibrium of the $n$-body problem