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On the chaotic diffusion in multidimensional Hamiltonian systems

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Abstract

We present numerical evidence that diffusion in the herein studied multidimensional near-integrable Hamiltonian systems departs from a normal process, at least for realistic timescales. Therefore, the derivation of a diffusion coefficient from a linear fit on the variance evolution of the unperturbed integrals fails. We review some topics on diffusion in the Arnold Hamiltonian and yield numerical and theoretical arguments to show that in the examples we considered, a standard coefficient would not provide a good estimation of the speed of diffusion. However, numerical experiments concerning diffusion would provide reliable information about the stability of the motion within chaotic regions of the phase space. In this direction, we present an extension of previous results concerning the dynamical structure of the Laplace resonance in Gliese-876 planetary system considering variations of the orbital parameters accordingly to the error introduced by the radial velocity determination. We found that a slight variation of the eccentricity of planet c would destabilize the inner region of the resonance that, though chaotic, shows stable when adopting the best fit values for the parameters.

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Notes

  1. We follow the same approach given in Cincotta and Giordano (2016).

  2. The generalization of a saddle equilibrium point, it is defined as the connected intersection of the stable and unstable manifolds or, in Arnold jargon, arriving and departing whiskers, \(W^-\) and \(W^+\), respectively [see Arnold (1964), Giorgilli (1990) for further details].

  3. Note that their intersections in action space are different.

  4. For a multiple resonance or an overlapped domain of resonances where there is not any specific direction \(I_f=\sqrt{I_1^2+I_2^2}\).

  5. See for instance Efthymiopoulos et al. (1999), Miguel et al. (2014) for the standard map in case of large values of the perturbation parameter.

  6. This nominal set is identical to those experiments labeled as 2–9 in MCB16.

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Acknowledgements

The authors were supported with grants from the Consejo de Investigaciones Científicas y Técnicas de la República Argentina (CONICET), the Universidad Nacional de La Plata (UNLP) and Universidad Nacional de Córdoba (UNC). This work used computational resources from CCAD UNC, in particular the Mendieta Cluster, which is part of SNCAD MinCyT, Argentina. Other numerical simulations were carried out on the local computing resources from the Instituto de Astronomía Teórica y Experimental (IATE), at the UNC, the Instituto de Astrofísica de La Plata (IALP) CONICET-UNLP and also on the IFLySIB computational resources at the Instituto de Física de Líquidos y Sistemas Biológicos, CONICET-UNLP. PMC would like to acknowledge C. Simó and A. Vieiro for their valuable comments, suggestions and discussions. The two anonymous reviewers are also acknowledged for their criticism that helped us to improve this manuscript.

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  1. Grupo de Caos en Sistemas Hamiltonianos, Facultad de Ciencias Astonómicas y Geofísicas, Universidad Nacional de La Plata and Instituto de Astrofísica de La Plata (CONICET-UNLP), La Plata, Argentina

    P. M. Cincotta, C. M. Giordano & J. G. Martí

  2. Instituto de Astronomía Teórica y Experimental (IATE), Observatorio Astronómico, Universidad Nacional de Córdoba, Córdoba, Argentina

    C. Beaugé

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  1. P. M. Cincotta
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  2. C. M. Giordano
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Cincotta, P.M., Giordano, C.M., Martí, J.G. et al. On the chaotic diffusion in multidimensional Hamiltonian systems. Celest Mech Dyn Astr 130, 7 (2018). https://doi.org/10.1007/s10569-017-9797-1

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