# Global instability in the restricted planar elliptic three body problem

Delshams A, Kaloshin V, de la Rosa A, Seara TM. 2018. Global instability in the restricted planar elliptic three body problem. Communications in Mathematical Physics. 366(3), 1173–1228.

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*Journal Article*|

*Published*|

*English*

Author

Delshams, Amadeu;
Kaloshin, Vadim

^{ISTA}^{}; de la Rosa, Abraham; Seara, Tere M.Abstract

The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet or an asteroid) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum. The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold P+ (resp. P−). In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity P∞, which we call the manifold at parabolic infinity. On P∞ it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. Since the inner dynamics inside P∞ is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion.

Publishing Year

Date Published

2018-09-05

Journal Title

Communications in Mathematical Physics

Volume

366

Issue

3

Page

1173-1228

IST-REx-ID

### Cite this

Delshams A, Kaloshin V, de la Rosa A, Seara TM. Global instability in the restricted planar elliptic three body problem.

*Communications in Mathematical Physics*. 2018;366(3):1173-1228. doi:10.1007/s00220-018-3248-zDelshams, A., Kaloshin, V., de la Rosa, A., & Seara, T. M. (2018). Global instability in the restricted planar elliptic three body problem.

*Communications in Mathematical Physics*. Springer Nature. https://doi.org/10.1007/s00220-018-3248-zDelshams, Amadeu, Vadim Kaloshin, Abraham de la Rosa, and Tere M. Seara. “Global Instability in the Restricted Planar Elliptic Three Body Problem.”

*Communications in Mathematical Physics*. Springer Nature, 2018. https://doi.org/10.1007/s00220-018-3248-z.A. Delshams, V. Kaloshin, A. de la Rosa, and T. M. Seara, “Global instability in the restricted planar elliptic three body problem,”

*Communications in Mathematical Physics*, vol. 366, no. 3. Springer Nature, pp. 1173–1228, 2018.Delshams, Amadeu, et al. “Global Instability in the Restricted Planar Elliptic Three Body Problem.”

*Communications in Mathematical Physics*, vol. 366, no. 3, Springer Nature, 2018, pp. 1173–228, doi:10.1007/s00220-018-3248-z.