Shannon entropy diffusion estimates: sensitivity on the parameters of the method

Abstract

In the present effort, we revisit the Shannon entropy approach for the study of both the extent and the rate of diffusion in a given dynamical system. In particular, we provide a theoretical and numerical study of the dependence of the formulation on the parameters of the method. We succeed in deriving not only a diffusion coefficient, \(D_{S}\), but also an estimate of the macroscopical instability time for the system under study. Dealing with a toy model, namely a 4D symplectic application that represents the dynamics around a junction of resonances of different order, and an a particular case of the planar three-body problem, the HD20003 planetary system, we obtain numerical evidence that \(D_{S}\) is a robust measure of the diffusion rate, no significant dependence on the free parameter of the entropy formulation (the size of the elements of the partition) being observed. Moreover, successful results concerning the estimation of macroscopical instability times obtained from \(D_{S}\) are presented in both cases.

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Acknowledgements

This work was supported by grants from Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina (CONICET), the Universidad Nacional de La Plata and the Universidad Nacional de Córdoba, IAG-USP and CAPES. We are very grateful to Carles Simó for his valuable comments that allow us to improve the manuscript. Two anonymous reviewers are also acknowledged for their criticism that helped us to improve this manuscript.

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Correspondence to Pablo M. Cincotta.

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Cincotta, P.M., Giordano, C.M., Silva, R.A. et al. Shannon entropy diffusion estimates: sensitivity on the parameters of the method. Celest Mech Dyn Astr 133, 7 (2021). https://doi.org/10.1007/s10569-021-10006-y

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Keywords

  • Chaotic diffusion
  • Resonances
  • Shannon entropy
  • Three-body problem