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Theory and applications of fast Lyapunov indicators to model problems of celestial mechanics

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Abstract

In the last decades, we have seen a rapid increment in the use of finite-time chaos indicators in celestial mechanics. They have been used to analyze the complex dynamics of planetary systems, of minor planets and of space debris. In fact, theoretical studies on fundamental dynamical models have revealed that, computed on short time intervals, they allow to efficiently detect resonances, represent the phase portraits of complex dynamics, compute center-stable-unstable manifolds as well as Lagrangian coherent structures. In this paper, we focus on applications of the fast Lyapunov indicator (FLI) and review through examples why its computation is particularly powerful for those systems whose solutions may have an asymptotic behavior very different from the short-term one, as it is the case of sequences of close encounters in gravitational systems and the advection of particles in aperiodic flows. The main case study here considered is the computation of the manifold tubes and the related transit orbits in the restricted three-body problem. We also provide a new application of the FLI to a complex problem of planetary hydrodynamics, such as the detection of the stable and unstable manifolds guiding the motions of particles advected by the gas of a protoplanetary nebula.

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Data Availability

The code fargoOCA is available at: https://disc.pages.oca.eu/fargOCA/public/.

Notes

  1. For the Sun–Jupiter mass ratio, we have \(C_1= 3.0387...\), \(C_2= 3.0374...\); for the numerical experiments we use \(C=3.0368...\). We remark that the values of the Jacobi constant \(C_1,C_2\) are very close, and in particular, small changes of C close to \(C_1,C_2\) determine important changes in the shape of the realm of possible motions.

  2. The definition depends on the local inversion map used to choose \(\zeta (0)\): the indicators (29) are therefore locally defined; as a matter of fact it is sufficient to use a couple of indicators \(\textrm{mFLI}^{\pm }_\phi (\xi ,w(0),T)\) to cover the phase space, see Guzzo and Lega (2018).

  3. http://www.openexoplanetcatalogue.com/.

  4. A recently re-factorized version of the code can be found at: https://disc.pages.oca.eu/fargOCA/public/. The code is parallelized using a hybrid combination of MPI between the nodes and accelerated using the Kokkos library. This gives the possibility of running the code on a large variety of machine’s architectures mixing CPUs and GPUs.

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Acknowledgements

We acknowledge HPC resources from GENCI DARI n.A0120407233 and from “Mesocentre SIGAMM” hosted by Observatoire de la Côte d’Azur. M.G. acknowledges the project MIUR-PRIN 20178CJA2B “New frontiers of Celestial Mechanics: theory and applications.”

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  1. Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova, Via Trieste 63, 35121, Padova, Italy

    Massimiliano Guzzo

  2. Laboratoire Lagrange, UMR7293, CNRS, Observatoire de la Côte d’Azur, Université de la Côte d’Azur, Boulevard de l’Observatoire, 06304, Nice Cedex 4, France

    Elena Lega

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Dedicated to the memory of Claude Froeschlé, we had the immense honor to embrace your intuition for disentangling intricate dynamics into beautiful and elegant images.

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Guzzo, M., Lega, E. Theory and applications of fast Lyapunov indicators to model problems of celestial mechanics. Celest Mech Dyn Astron 135, 37 (2023). https://doi.org/10.1007/s10569-023-10152-5

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