Shannon entropy applied to the planar restricted three-body problem

  • C. BeaugéEmail author
  • P. M. Cincotta
Original Article


We present a numerical study of the application of the Shannon entropy technique to the planar restricted three-body problem in the vicinity of first-order interior mean-motion resonances with the perturber. We estimate the diffusion coefficient for a series of initial conditions and compare the results with calculations obtained from the time evolution of the variance in the semimajor axis and eccentricity plane. Adopting adequate normalization factors, both methods yield comparable results, although much shorter integration times are required for entropy calculations. A second advantage of the use of entropy is that it is possible to obtain reliable results even without the use of ensembles or analysis restricted to surfaces of section or representative planes. This allows for a much more numerically efficient tool that may be incorporated into a working N-body code and applied to numerous dynamical problems in planetary dynamics. Finally, we estimate instability times for a series of initial conditions in the 2/1 and 3/2 mean-motion resonances and compare them with times of escape obtained from directed N-body simulations. We find very good agreement in all cases, not only with respect to average values but also in their dispersion for nearby trajectories.


Three-body problem Resonances Stability 



Most of the calculations necessary for this work were carried out with the computing facilities of IATE/UNC as well as in the High Performance Computing Center of the Universidad Nacional de Córdoba (CCAD-UNC). This research was funded by CONICET, Secyt/UNC and FONCYT.

Compliance with ethical standards

Conflicts of interest

The authors have no conflict of interest to declare.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Instituto de Astronomía Teórica y Experimental (IATE), Observatorio AstronómicoUniversidad Nacional de CórdobaCordobaArgentina
  2. 2.Grupo de Caos en Sistemas Hamiltonianos, Facultad de Ciencias Astronómicas y GeofísicasUniversidad Nacional de La Plata and Instituto de Astrofísica de La Plata (CONICET-UNLP)Buenos AiresArgentina

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