Modifications of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) linear variational method called MEGNO2 and OMEGNO2 indicators are introduced. The modifications are based on taking into account not only the linear, but also the nonlinear part of the increment of the phase flow in the divergence among nearby trajectories according to the second-order formulas. The new indicators allow one to determine more quickly the nature of the orbits under study in dynamical systems with zero or small Lyapunov exponents in comparison with the first-order variational indicators. They improve the analysis of regular regions and, in particular, periodic orbits as well as prevent the appearance of spurious structures in the resulting mappings.
Similar content being viewed by others
References
C. Froeschlé, E. Lega, and R. Gonczi, Celest. Mech. Dyn. Astr., 67, 41–62 (1997).
C. Froeschlé and E. Lega, Celest. Mech. Dyn. Astr., 78, 167–195 (2000).
N. Voglis, G. Contopoulos, and C. Efthymiopoulos, Celest. Mech. Dyn. Astr., 37, 211–220 (1999).
P. M. Cincotta and C. Simo, Astron. Astrophys. Sup., 147, 205–228 (2000).
P. M. Cincotta, C. M. Giordano, and C. Simo, Physica D, 182, 151–178 (2003).
Ch. Skokos, J. Phys. A, 34, 10029–10043 (2001).
Ch. Skokos, T. C. Bountis, and Ch. Antonopoulos, Physica D, 231, 30–54 (2007).
G. Lukes-Gerakopoulos, N. Voglis, and C. Efthymiopoulos, Physica A, 387, 1907–1925 (2008).
N. P. Maffione et al., Celest. Mech. Dyn. Astr., 111, 285–307 (2011).
L. A. Darriba et al., Int. J. Bifurcation and Chaos, 22, 1230033 (2012).
A. M. Koksin and V. A. Shefer, Proc. IAU Symp., 9, No. 310, 35–38 (2014).
E. Lega and C. Froeschlé, Celest. Mech. Dyn. Astr., 81, 129–147 (2001).
M. Fouchard et al., Celest. Mech. Dyn. Astr., 83, 205–222 (2002).
R. Barrio, Chaos, Solutions and Fractals, 25, 711–726 (2005).
K. Okubo and K. Umeno, Preprint, arXiv:1501.00243v5 (18 March 2015).
V. A. Shefer and A. M. Koksin, Izv. Vyssh. Uchebn. Zaved. Fiz., 56, No. 6/3, 256–258 (2013).
Ch. Skokos, Lect. Notes Phys., 790, 63–135 (2010).
P. M. Cincotta and C. M. Giordano, Lect. Notes Phys., 915, 93–128 (2016).
K. Goździewski et al., Astron. Astrophys., 378, 569–586 (2001).
S. Valk et al., Adv. Space Res., 43, 1509–1526 (2009).
T. C. Hinse et al., Mon. Not. R. Astron. Soc., 404, 837–857 (2010).
G. Benettin, L. Galgani, and J.-M. Strelcyn, Phys. Rev. A, 14, 2338–2345 (1976).
T. C. Hinse et al., Astron. Astrophys., 488, 1133–1147 (2008).
V. A. Shefer, Izv. Vyssh. Uchebn. Zaved. Fiz., 54, No. 6/2, 13–21 (2011).
R. Barrio, W. Borczyk, and S. Breiter, Chaos, Solutions and Fractals, 40, 1697–1714 (2009).
M. Guzzo, E. Lega, and C. Froeschlé, Physica D, 163, 1–25 (2002).
V. A. Shefer and A. M. Koksin, Russ. Phys. J., 59, No. 1, 71–75 (2016).
R. Barrio, Lect. Notes Phys., 915, 55–92 (2016).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 71–79, October, 2017.
Rights and permissions
About this article
Cite this article
Shefer, V.A. Second-Order Chaos Indicators MEGNO2 and OMEGNO2: Theory. Russ Phys J 60, 1728–1738 (2018). https://doi.org/10.1007/s11182-018-1275-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11182-018-1275-z