Numerical Continuation of Hamiltonian Relative Periodic Orbits |
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Authors: | Claudia Wulff Andreas Schebesch |
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Institution: | 1. Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK 2. Fachbereich Mathematik und Informatik, Freie Universit?t Berlin, 14195, Berlin, Germany
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Abstract: | The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years,
there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry
or symplecticity. But as yet, there are few results on the numerical computation of those bifurcations. The methods we present
in this paper are a first step toward a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic
orbits and relative periodic orbits (RPOs). First, we show how to numerically exploit spatio-temporal symmetries of Hamiltonian
periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in
a symmetry breaking bifurcation. Finally, we present an algorithm for the numerical continuation of non-degenerate Hamiltonian
relative periodic orbits with regular drift-momentum pair. Our path following algorithm is based on a multiple shooting algorithm
for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with
implicit reparametrization. We apply our methods to continue the famous figure eight choreography of the three-body system.
We find a relative period doubling bifurcation of the planar rotating eight family and compute the rotating choreographies
bifurcating from it.
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Keywords: | Numerical continuation Symmetric Hamiltonian systems Relative periodic orbits |
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