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Adjusting chaotic indicators to curved spacetimes

Georgios Lukes-Gerakopoulos*

  • Theoretical Physics Institute, University of Jena, 07743 Jena, Germany

  • *gglukes@gmail.com
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Phys. Rev. D 89, 043002 – Published 5 February, 2014

DOI: https://doi.org/10.1103/PhysRevD.89.043002

Abstract

In this work, chaotic indicators, which have been established in the framework of classical mechanics, are reformulated in the framework of general relativity in such a way that they are invariant under coordinate transformation. For achieving this, the prescription for reformulating mLCE given by [Y. Sota, S. Suzuki, and K.-I. Maeda, Classical Quantum Gravity 13, 1241 (1996)] is adopted. Thus, the geodesic deviation vector approach is applied, and the proper time is utilized as the measure of time. Following the aforementioned prescription, the chaotic indicators FLI, MEGNO, GALI, and APLE are reformulated. In fact, FLI has been reformulated by adapting other prescriptions in the past, but not by adapting the Sota et al. one. By using one of these previous reformulations of FLI, an approximative expression giving MEGNO as function of FLI has been applied on nonintegrable curved spacetimes in a recent work. In the present work the reformulation of MEGNO is provided by adjusting the definition of the indicator to the Sota et al. prescription. GALI and APLE are reformulated in the framework of general relativity for the first time. All the reformulated indicators by Sota et al. prescription are tested and compared for their efficiency to discern order from chaos.

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References (36)

  1. R. L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley Publishing Company, New York, 1989).
  2. J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, Am. Math. Mon. 99, 332 (1992).
  3. G. Contopoulos, Order and Chaos in Dynamical Astronomy (Springer, Berlin, 2002).
  4. C. Skokos, Lect. Notes Phys. 790, 63 (2010).
  5. N. P. Maffione, L. A. Darriba, P. M. Cincotta, and C. M. Giordano, Celest. Mech. Dyn. Astron. 111, 285 (2011); Mon. Not. R. Astron. Soc. 429, 2700 (2013).
  6. C. Froeschlé, R. Gonczi, and E. Lega, Planet. Space Sci. 45, 881 (1997).
  7. C. Froeschlé, M. Guzzo, and E. Lega, Celest. Mech. Dyn. Astron. 92, 243 (2005).
  8. P. M. Cincotta and C. Simó, Astron. Astrophys. Suppl. Ser. 147, 205 (2000).
  9. P. M. Cincotta, C. M. Giordano, and C. Simó, Physica (Amsterdam) 182D, 151 (2003).
  10. C. Skokos, J. Phys. A 34, 10029 (2001).
  11. C. Skokos, T. C. Bountis, and C. Antonopoulos, Physica (Amsterdam) 231D, 30 (2007).
  12. G. Lukes-Gerakopoulos, N. Voglis, and C. Efthymiopoulos, Physica (Amsterdam) 387A, 1907 (2008).
  13. G. Lukes-Gerakopoulos, N. Voglis, and C. Efthymiopoulos, Chaos in Astronomy, Astrophysics and Space Science Proceedings (Springer, New York, 2009), p. 363.
  14. A. E. Motter, Phys. Rev. Lett. 91, 231101 (2003); K. Gelfert and A. E. Motter, Commun. Math. Phys. 300, 411 (2010).
  15. Y. Sota, S. Suzuki, and K.-I. Maeda, Classical Quantum Gravity 13, 1241 (1996).
  16. X. Wu, T.-Y. Huang, and H. Zhang, Phys. Rev. D 74, 083001 (2006).
  17. V. Karas and D. Vokrouhlicky, Gen. Relativ. Gravit. 24, 729 (1992).
  18. M. D. Hartl, Phys. Rev. D 67, 024005 (2003); 67, 104023 (2003).
  19. W. Han, Gen. Relativ. Gravit. 40, 1831 (2008).
  20. J. Laskar, Celest. Mech. Dyn. Astron. 56, 191 (1993).
  21. O. Semerák and P. Suková, Mon. Not. R. Astron. Soc. 404, 545 (2010); 425, 2455 (2012); P. Suková and O. Semerák, ibid. 436, 978 (2013).
  22. G. Lukes-Gerakopoulos, Phys. Rev. D 86, 044013 (2012).
  23. N. Marwan, M. Carmen Romano, M. Thiel, and J. Kurths, Phys. Rep. 438, 237 (2007).
  24. O. Kopáček, V. Karas, J. Kovář, and Z. Stuchlík, Astrophys. J. 722, 1240 (2010); J. Kovář, O. Kopáček, V. Karas, and Y. Kojima, Classical Quantum Gravity 30, 025010 (2013).
  25. C. P. Dettmann, N. E. Frankel, and N. J. Cornish, Phys. Rev. D 50, R618 (1994); N. J. Cornish and N. E. Frankel, ibid. 56, 1903 (1997); N. J. Cornish and J. Levin, ibid. 68, 024004 (2003).
  26. M. Szydłowski, Phys. Lett. A 176, 22 (1993); M. Szydłowski and A. Krawiec, Phys. Rev. D 53, 6893 (1996).
  27. V. S. Manko, J. D. Sanabria-Gómez, and O. V. Manko, Phys. Rev. D 62, 044048 (2000).
  28. V. S. Manko, J. Martín, and E. Ruiz, Phys. Rev. D 51, 4187 (1995).
  29. G. Pappas and T. A. Apostolatos, Mon. Not. R. Astron. Soc. 429, 3007 (2013).
  30. E. Berti and N. Stergioulas, Mon. Not. R. Astron. Soc. 350, 1416 (2004).
  31. F. L. Dubeibe, L. A. Pachón, and J. D. Sanabria-Gómez, Phys. Rev. D 75, 023008 (2007).
  32. W.-B. Han, Phys. Rev. D 77, 123007 (2008).
  33. J. Seyrich and G. Lukes-Gerakopoulos, Phys. Rev. D 86, 124013 (2012).
  34. M. F. Mestre, P. M. Cincotta, and C. M. Giordano, Mon. Not. R. Astron. Soc. 414, L100 (2011).
  35. N. Voglis, C. Kalapotharakos, and I. Stavropoulos, Mon. Not. R. Astron. Soc. 337, 619 (2002).
  36. N. Voglis, G. Contopoulos, and C. Efthymiopoulos, Phys. Rev. E 57, 372 (1998); Celest. Mech. Dyn. Astron. 73, 211 (1999).

Outline

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Phys. Rev. D 89, 043002– Published 5 February, 2014
  • Received 26 November 2013

© 2014 American Physical Society

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