Chaotic quasi-collision trajectories in the 3-centre problem

TL;DR

This paper demonstrates the existence of chaotic and hyperbolic trajectories in the planar 3-centre problem for small perturbations, without assuming the third centre's position, using regularisation and invariant set construction.

Contribution

It extends previous results by removing positional assumptions on the third centre and proves chaos in a more general setting with small perturbations.

Findings

Existence of hyperbolic invariant sets for a dense set of third centre positions.

Construction of collision arc chains via regularisation techniques.

Chaotic almost collision orbits are proven to exist.

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. Diff. Eq. 190 (2003), 539--558): this result has been obtained by the use of the Poincar\'e-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 on the real plane, 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 (see Cel. Mech. & Dyn. Astr. 77 (2000), 49--75). 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.