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


The predictability of a system aims to characterise if a numerically computed orbit may be sometimes sufficiently close to another true solution, so it may be still reflecting real properties of the model, leading to correct predictions. The real orbit is called a shadow, and the shadowing property characterises the validity of long computer simulations, and how they may be globally sensitive to small errors. We can derive a predictability index from the computation of the distributions of finite-time Lyapunov exponents. For doing so, we need to estimate the most suitable finite-time interval size, which is associated with the time scales when the flow leaves the local regime and any initial perturbation begins to converge towards the globally most fasting growing direction.


Lyapunov Exponent Interval Length Interval Size Asymptotic Regime Chaotic Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  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.
    Athanassoula, E., Romero-Gómez, M., Bosma, A., Masdemont, J.J.: Rings and spirals in barred galaxies - III. Further comparisons and links to observations. Mon. Not. R. Astron. Soc. 407, 1433 (2010)Google Scholar
  3. 3.
    Buljan, H., Paar, V.: Many-hole interactions and the average lifetimes of chaotic transients that precede controlled periodic motion. Phys. Rev. E 63, 066205 (2001)ADSCrossRefGoogle Scholar
  4. 4.
    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)ADSMathSciNetCrossRefzbMATHGoogle 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, Hauppauge (2008)Google Scholar
  8. 8.
    Contopoulos, G.: Orbits in highly perturbed dynamical systems. I. Periodic orbits. Astron. J. 75, 96 (1970)ADSMathSciNetGoogle Scholar
  9. 9.
    Contopoulos, G., Harsoula, M.: 3D chaotic diffusion in barred spiral galaxies. Mon. Not. R. Astron. Soc. 436, 1201 (2013)ADSCrossRefGoogle Scholar
  10. 10.
    Contopoulos, G., Grousousakou, E., Voglis, N.: Invariant spectra in Hamiltonian systems. Astron. Astrophys. 304, 374 (1995)ADSGoogle Scholar
  11. 11.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Do, Y., Lai, Y.C.: Statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability. Phys. Rev. E 69, 16213 (2004)ADSCrossRefGoogle Scholar
  13. 13.
    Grassberger, P., Badii, R., Politi, A.: Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51, 135 (1988)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations, I, Nonstiff Problems, 2nd edn. Springer, Berlin (1993)zbMATHGoogle Scholar
  15. 15.
    Johnston, K.V., Spergel, D.N., Hernquist, L.: The disruption of the sagittarius dwarf galaxy. Astrophys. J. 451, 598 (1995)ADSCrossRefGoogle Scholar
  16. 16.
    Kapitaniak, T.: Distribution of transient Lyapunov exponents of quasiperiodically forced systems. Prog. Theor. Phys. 93, 831 (1995)ADSCrossRefGoogle Scholar
  17. 17.
    Kottos, T., Politi, A., Izrailev, F.M., Ruffo, S.: Scaling properties of Lyapunov spectra for the band random matrix model. Phys. Rev. E 53, 6 (1996)CrossRefGoogle Scholar
  18. 18.
    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
  19. 19.
    Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. Applied Mathematical Sciences, vol. 38, 2nd edn. Springer, New York (1992)Google Scholar
  20. 20.
    Maffione, N.P., Darriba, L.A., Cincotta, P.M., Giordano, C.M.: Chaos detection tools: application to a self-consistent triaxial model. Mon. Not. R. Astron. Soc. 429, 2700 (2013)ADSCrossRefGoogle Scholar
  21. 21.
    Manos, T., Athanassoula, E.: Regular and chaotic orbits in barred galaxies - I. Applying the SALI/GALI method to explore their distribution in several models. Mon. Not. R. Astron. Soc. 415, 629 (2011)Google Scholar
  22. 22.
    Manos, T., Machado, R.E.G.: Chaos and dynamical trends in barred galaxies: bridging the gap between N-body simulations and time-dependent analytical models. Mon. Not. R. Astron. Soc. 438, 2201 (2014)ADSCrossRefGoogle Scholar
  23. 23.
    Miyamoto, M., Nagai, R.: Three dimensional models for the distribution of mass in galaxies. Publ. Astron. Soc. Jpn. 27, 533 (1975)ADSGoogle Scholar
  24. 24.
    Pfenniger, D.: The 3D dynamics of barred galaxies. Astron. Astrophys. 134, 373 (1984)ADSMathSciNetGoogle Scholar
  25. 25.
    Prasad, A., Ramaswany, R.: Characteristic distributions of finite-time Lyapunov exponents. Phys. Rev. E 60, 2761 (1999)ADSCrossRefGoogle Scholar
  26. 26.
    Sauer, T.: Shadowing breakdown and large errors in dynamical simulations of physical systems. Phys. Rev. E. 65, 036220 (2002)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Sauer, T., Grebogi, C., Yorke, J.A.: How long do numerical chaotic solutions remain valid? Phys. Lett. A 79, 59 (1997)CrossRefGoogle Scholar
  28. 28.
    Sepulveda, M.A., Badii, R., Pollak, E.: Spectral analysis of conservative dynamical systems. Phys. Lett. 63, 1226 (1989)CrossRefGoogle Scholar
  29. 29.
    Skokos, Ch., 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
  30. 30.
    Tomsovic, S., Lakshminarayan, A.: Fluctuations of finite-time stability exponents in the standard map and the detection of small islands. Phys. Rev. E 76, 036207 (2007)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Tsiganis, K., Varvoglis, H., Hadjidemetriou, J.D.: Stable chaos in high-order Jovian resonances. Icarus, 155, 454 (2002)ADSCrossRefGoogle Scholar
  32. 32.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Vallejo, J.C., Viana, R., Sanjuan, M.A.F.: Local predictability and non hyperbolicity through finite Lyapunov Exponents distributions in two-degrees-of-freedom Hamiltonian systems. Phys. Rev. E 78, 066204 (2008)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Vallejo, J.C., Sanjuan, M.A.F.: Predictability of orbits in coupled systems through finite-time Lyapunov exponents. N. J. Phys. 15, 113064 (2013)CrossRefGoogle Scholar
  35. 35.
    Viana, R.L., Pinto, S.E., Barbosa, J.R., Grebogi, C.: Pseudo-deterministic chaotic systems. Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, 1 (2003)Google Scholar
  36. 36.
    Viana, R.L., Barbosa, J.R., Grebogi, C., Batista, C.M.: Simulating a chaotic process. Braz. J. Phys. 35, 1 (2005)ADSCrossRefGoogle Scholar
  37. 37.
    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
  38. 38.
    Westfall, P.H.: Kurtosis as peakedness, 1905–2014, R.I.P. Am. Stat. 68, 191 (2014)Google Scholar
  39. 39.
    Yanchuk, S., Kapitaniak, T.: Symmetry increasing bifurcation as a predictor of chaos-hyperchaos transition in coupled systems. Phys. Rev. E 64, 056235 (2001)ADSCrossRefGoogle Scholar
  40. 40.
    Yanchuk, S., Kapitaniak, T.: Chaos-hyperchaos transition in coupled Rössler systems. Phys. Lett. A 290, 139 (2001)ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Juan C. Vallejo
    • 1
  • Miguel A. F. Sanjuan
    • 1
  1. 1.Department of PhysicsUniversidad Rey Juan CarlosMóstoles, MadridSpain

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