Chaotic quasi-collision trajectories in the 3-centre problem |
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Authors: | Linda Dimare |
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Institution: | (2) CMUP, Porto, Portugal;(3) IM-UFRJ, Rio de Janeiro, Brazil;(4) IMPA, Rio de Janeiro, Brazil; |
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Abstract: | We study a particular kind of chaotic dynamics for the planar 3-centre problem on small negative energy level sets. We know
that chaotic motions exist, if we make the assumption that one of the centres is far away from the other two (see Bolotin
and Negrini, J Differ Equ 190:539–558, 2003): this result has been obtained by the use of the Poincaré-Melnikov theory. Here
we change the assumption on the third centre: we do not make any hypothesis on its position, and we obtain a perturbation
of the 2-centre problem by assuming its intensity to be very small. Then, for a dense subset of possible positions of the
perturbing centre in
\mathbbR2{\mathbb{R}^2} , we prove the existence of uniformly hyperbolic invariant sets of periodic and chaotic almost collision orbits by the use
of a general result of Bolotin and MacKay (Celest Mech Dyn Astron 77:49–75, 77:49–75, 2000; Celest Mech Dyn Astron 94(4):433–449,
2006). To apply it, we must preliminarily construct chains of collision arcs in a proper way. We succeed in doing that by the classical regularisation of the 2-centre problem and the use of the periodic
orbits of the regularised problem passing through the third centre. |
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Keywords: | |
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