A Detailed Example: Galactic Dynamics

  • Juan C. Vallejo
  • Miguel A. F. Sanjuan
Part of the Springer Series in Synergetics book series (SSSYN)


We have seen the impact of the presence of chaos in a given dynamical system. Furthermore, the analysis of the finite-time Lyapunov exponent distributions allows the computation of the predictability indicator, which characterises the presence of shadowing in a given orbit.


  1. 1.
    Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems, p. 383. Springer, New York (1996)Google Scholar
  2. 2.
    Cachucho, F., Cincotta, P.M., Ferraz-Mello, S.: Chirikov diffusion in the asteroidal three-body resonance (5, -2, -2). Celest. Mech. Dyn. Astron. 108, 35 (2010)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Caranicolas, N.D., Zotos, E.E.: The evolution of chaos in active galaxy models with an oblate or a prolate dark halo component. Astron. Nachr. 331, 330 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    Carpintero, D., Muzzio, J.C., Navone, H.D.: Models of cupsy triaxial stellar systems – III. The effect of velocity anisotropy on chaoticity. Mon. Not. R. Astron. Soc. 438, 287 (2014)Google Scholar
  5. 5.
    Casertano, S., Ratnatunga, K.U., Bahcalli, J.N.: Kinematic modeling of the galaxy. II – two samples of high proper motion stars. Astrophys. J. 357, 435 (1990)Google Scholar
  6. 6.
    Chiba, M., Beers, T.C.: Structure of the galactic stellar halo prior to disk formation. Astrophys. J. 549, 325 (2001)ADSCrossRefGoogle Scholar
  7. 7.
    Cincotta, P.M., Giordano, C.M.: Topics on diffusion in phase space of multidimensional Hamiltonian systems. In: New Nonlinear Phenomena Research, p. 319. Nova Science Publishers, Inc., New York (2008)Google Scholar
  8. 8.
    Contopoulos, G., Harsoula, M.: 3D chaotic diffusion in barred spiral galaxies. Mon. Not. R. Astron. Soc. 436, 1201 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    Davidchack, R.L., Lai, Y.C.: Characterization of transition to chaos with multiple positive Lyapunov exponents by unstable periodic orbits. Phys. Lett. A 270, 308 (2000)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Habib, T., Kandrup, H.E., Mahon, M.E.: Chaos and noise in Galactic potentials. Astrophys. J. 480, 155 (1997)ADSCrossRefGoogle Scholar
  11. 11.
    Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations I: nonstiff problems, 2nd ed. Springer, Berlin (1993)zbMATHGoogle Scholar
  12. 12.
    Hayes, W.B.: Shadowing-based reliability decay in softened n-body simulations. Astrophys. J. 587, L59–L62 (2003)ADSCrossRefGoogle Scholar
  13. 13.
    Johnston, K.V., Spergel, D.N., Hernquist, L.: The disruption of the sagittarius dwarf galaxy. Astrophys. J. 451, 598 (1995)ADSCrossRefGoogle Scholar
  14. 14.
    Kantz, H., Grebogi, C., Prasad, A., Lai, Y.C., Sinde, E.: Unexpected robustness-against-noise of a class of nonhyperbolic chaotic attractors. Phys. Rev. E, 65, 026209 (2002)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Law, D.R., Majewski, S.R., Johnston, K.V.: Evidence for a triaxial Milky Way dark matter halo from the sagittarius stellar tidal stream. Astrophys. J. Lett. 703, L67 (2009)ADSCrossRefGoogle Scholar
  16. 16.
    Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics, 2nd ed. Applied Mathematical Sciences, vol. 38. Springer, New York (1992)Google Scholar
  17. 17.
    Martinez-Valpuesta, I., Shlosman, I.: Why buckling stellar bars weaken in disk galaxies. Astrophys. J. 613, 613L (2004)CrossRefGoogle Scholar
  18. 18.
    Miyamoto, M., Nagai, R.: Three dimensional models for the distribution of mass in galaxies. Publ. Astron. Soc. Jpn. 27, 533 (1975)ADSGoogle Scholar
  19. 19.
    Olle, M., Pfenigger, D.: Vertical orbital structure around the Lagrangian points in barred galaxies. Link with the secular evolution of galaxies. Astron. Astrophys. 334, 829 (1998)Google Scholar
  20. 20.
    Papaphilippou, Y., Laskar, J.: Global dynamics of triaxial galactic models through frequency map analysis. Astron. Astrophys. 329, 451 (1998)ADSGoogle Scholar
  21. 21.
    Pfenniger, D.: The 3D dynamics of barred galaxies. Astron. Astrophys. 134, 373 (1984)ADSMathSciNetGoogle Scholar
  22. 22.
    Pfenniger, D.: Relaxation and dynamical friction in non-integrable stellar systems. Astron. Astrophys. 165, 74 (1986)ADSzbMATHGoogle Scholar
  23. 23.
    Pfenniger, D.: Dissipation in barred galaxies: the growth of bulges and central mass concentrations. Astrophys. J. 363, 391 (1990)ADSCrossRefGoogle Scholar
  24. 24.
    Pfenniger, D., Firedli, D.: Structure and dynamics of 3D N-body barred galaxies. Astron. Astrophys. 252, 75 (1991)ADSGoogle Scholar
  25. 25.
    Skokos, C., Patsis, P.A., Athanassoula, E.: Orbital dynamics of three-dimensional bars – I. The backbone of three-dimensional bars. A fiducial case. Mon. Not. R. Astron. Soc. 333, 847 (2002)Google Scholar
  26. 26.
    Tsiganis, K., Varvoglis, H., Hadjidemetriou, J.D.: Stable chaos in high-order Jovian resonances. Icarus 155, 454 (2002)ADSCrossRefGoogle Scholar
  27. 27.
    Udry, S., Pfenniger, D.: Stochasticity in elliptical galaxies. Astron. Astrophys. 198, 135 (1988)ADSMathSciNetzbMATHGoogle Scholar
  28. 28.
    Vallejo, J.C., Sanjuan, M.A.F.: Predictability of orbits in coupled systems through finite-time Lyapunov exponents. New J. Phys. 15 113064 (2013)ADSCrossRefGoogle Scholar
  29. 29.
    Vallejo, J.C., Sanjuan, M.A.F.: The forecast of predictability for computed orbits in galactic models. Mon. Not. R. Astron. Soc. 447, 3797 (2015)ADSCrossRefGoogle Scholar
  30. 30.
    Vallejo, J.C., Sanjuan, M.A.F.: Role of dark matter haloes on the predictability of computed orbits. Astron. Astrophys. 595, A68 (2016)ADSCrossRefGoogle Scholar
  31. 31.
    Vallejo, J.C., Aguirre, J., Sanjuan M.A.F.: Characterization of the local instability in the Henon-Heiles Hamiltonian. Phys. Lett. A 311, 26 (2003)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Vallejo, J.C., Viana, R., Sanjuan, M.A.F.: Local predictibility and non hyperbolicity through finite Lyapunov exponents distributions in two-degrees-of-freedom Hamiltonian systems. Phys. Rev. E 78, 066204 (2008)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Viana, R.L., Grebogi, C.: Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462 (2000)ADSCrossRefGoogle Scholar
  34. 34.
    Viana, R.L., Barbosa, J.R., Grebogi, C., Batista, C.M.: Simulating a chaotic process. Braz. J. Phys. 35, 1 (2005)ADSCrossRefGoogle Scholar
  35. 35.
    Wang, Y., Zhao, H., Mao, S., Rich, R.M.: A new model for the Milky Way bar. Mon. Not. R. Astron. Soc. 427, 1429 (2012)ADSCrossRefGoogle Scholar
  36. 36.
    El-Zant, A.A., Hassler, B.: Dynamics of galaxies with triaxial haloes. New Astron. 3, 493 (1998)ADSCrossRefGoogle Scholar
  37. 37.
    El-Zant, A.A., Shlosman, I.: Dark Halo shapes and the fate of stellar bars. Astrophys. J. 577, 626 (2002)ADSCrossRefGoogle Scholar
  38. 38.
    Zotos, E.E.: Classifying orbits in galaxy models with a prolate or an oblate dark matter halo. Astron. Astrophys. 563, 19 (2014)ADSCrossRefGoogle Scholar
  39. 39.
    Zotos, E.E., Caranicolas, N.D.: Revealing the influence of dark matter on the nature of motion and the families of orbits in axisymmetric galaxy models. Astron. Astrophys. 560, 110 (2013)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Juan C. Vallejo
    • 1
  • Miguel A. F. Sanjuan
    • 1
  1. 1.Departamento de FisicaUniversidad Rey Juan CarlosMóstolesSpain

Personalised recommendations