Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the <Emphasis Type="Italic">n</Emphasis>-body problem |
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| Authors: | Renato Calleja Eusebius Doedel Carlos García-Azpeitia |
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| Institution: | 1.Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas,Universidad Nacional Autónoma de México,Mexico,Mexico;2.Department of Computer Science,Concordia University,Montreal,Canada;3.Facultad de Ciencias,Universidad Nacional Autónoma de México,Mexico,Mexico |
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| Abstract: | We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits and subsequently bifurcating families. The Lyapunov families arise from the polygonal equilibrium 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 \(n=3\), 4, and 6–9. 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 below. |
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