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The Relative Lyapunov Indicators: Theory and Application to Dynamical Astronomy

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Chaos Detection and Predictability

Part of the book series: Lecture Notes in Physics ((LNP,volume 915))

Abstract

A recently introduced chaos detection method, the Relative Lyapunov Indicator (RLI) is investigated in the cases of symplectic mappings and continuous Hamiltonian systems. It is shown that the RLI is an efficient numerical tool in determining the true nature of individual orbits, and in separating ordered and chaotic regions of the phase space of dynamical systems. A comparison between the RLI and some other variational indicators are presented, as well as the recent applications of the RLI to various problems of dynamical astronomy.

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Notes

  1. 1.

    In the case of continuous dynamical systems the trajectory is a continuous curve in the phase space given by the points representing the time evolution of an initial state. In discrete dynamical systems the set of the discrete points representing the time evolution of the system is called as orbit.

  2. 2.

    The values for the parameter have been taken from the interval suggested in Sect. 6.1.2.

  3. 3.

    The E norm is the normalized energy: \(\left (E - E_{mb}\right )/\left (E_{lb} - E_{mb}\right )\).

  4. 4.

    Remember that the integration time varies with the E norm , which explains the transition from dark to light gray in the background of the plots on the left side of Figs. 6.11 and 6.12. Also the time of integration is fixed to 0 where there are not initial conditions.

  5. 5.

    Here, the LIs are the numerical approximations of the spectra of Lyapunov Characteristic Exponents.

  6. 6.

    Further information on the LP-VIcode can be found at the following url: http://www.fcaglp.unlp.edu.ar/LP-VIcode/.

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Acknowledgements

The authors thank the invitation and support of the scientific coordinators of the international workshop on Methods of Chaos Detection and Predictability: Theory and Applications: Georg Gottwald, Jacques Laskar and Haris Skokos and the hospitality of the Max Planck Institute for the Physics of Complex Systems where the meeting took place. ZsS is supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. NM is supported with grants from the Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina (CCT - La Plata) and the Universidad Nacional de La Plata.

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Authors and Affiliations

  1. Computational Astrophysics Group, Konkoly Observatory of the Hungarian Academy of Sciences, Budapest, Hungary

    Zsolt Sándor

  2. Grupo de Caos en Sistemas Hamiltonianos, Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de la Plata, Buenos Aires, Argentina

    Nicolás Maffione

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  1. Zsolt Sándor
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  2. Nicolás Maffione
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Correspondence to Zsolt Sándor .

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Editors and Affiliations

  1. University of Cape Town, Department of Mathematics and Applied Ma, Rondebosch, South Africa

    Charalampos (Haris) Skokos

  2. School of Mathematics and Statistics, University of Sydney, Sydney, Australia

    Georg A. Gottwald

  3. Observatoire de Paris, IMCCE, Paris, France

    Jacques Laskar

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Sándor, Z., Maffione, N. (2016). The Relative Lyapunov Indicators: Theory and Application to Dynamical Astronomy. In: Skokos, C., Gottwald, G., Laskar, J. (eds) Chaos Detection and Predictability. Lecture Notes in Physics, vol 915. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48410-4_6

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