Advertisement

Dynamical Regimes and Timescales

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

Abstract

There is a variety of relevant timescales when dealing with Lyapunov exponents.

References

  1. 1.
    Abraham, R., Smale S.: Non-genericity of Ω-stability. Proc. Symp. Pure Math. 14, 5 (1970)CrossRefGoogle Scholar
  2. 2.
    Aguirre, J., Vallejo, J.C., Sanjuán, M.A.F.: Wada basins and chaotic invariant sets in the Hénon-Heiles system. Phys. Rev. E 64, 66208 (2001)ADSCrossRefGoogle Scholar
  3. 3.
    Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems, p. 383. Springer, New York (1996)Google Scholar
  4. 4.
    Alligood, K.T., Sander, E., Yorke, J.A.: Three-dimensional crisis: crossing bifurcations and unstable dimension variability. Phys. Rev. Lett. 96, 244103 (2006)ADSCrossRefGoogle Scholar
  5. 5.
    Barreto, E., So, P.: Mechanisms for the development of unstable dimension variability and the breakdown of shadowing in coupled chaotic systems. Phys. Rev. Lett. 85 2490 (2000)ADSCrossRefGoogle Scholar
  6. 6.
    Benzi, R., Parisi, G., Vulpiani, A.: Characterisation of intermittency in chaotic systems. J. Phys. A, 18, 2157 (1985)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press, Princeton (1987)zbMATHGoogle Scholar
  8. 8.
    Branicki, M., Wiggings, S.: Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time exponents. Nonlinear Process. Geophys. 17 (2010)Google Scholar
  9. 9.
    Contopoulos, G.: Orbits in highly perturbed dynamical systems. I. Periodic orbits. Astron. J. 75, 96 (1970)ADSMathSciNetGoogle 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)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Dawson, S.P., Grebogi, C., Sauer, T., Yorke, J.A.: Obstructions to shadowing when a Lyapunov exponent fluctuates about zero. Phys. Rev. Lett. 73, 1927 (1994)ADSCrossRefGoogle Scholar
  13. 13.
    Grassberger, P.: Generalizations of the Hausdorff dimension of fractal measures. Phys. Lett. A 107, 101 (1985)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Grassberger, P., Badii, R., Politi, A.: Scaling Laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51, 135 (1988)ADSCrossRefGoogle Scholar
  15. 15.
    Grebogi, C., Ott, E., Yorke, J.A.: Crises, sudden changes in chaotic attractors, and transient chaos. Phys. D 7, 181 (1983)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jacobs, J., Ott, E., Hunt, R.: Scaling of the durations of chaotic transients in windows of attracting periodicity. Phys. Rev. E 56, 6508 (1997)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    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
  19. 19.
    Lai, Y.C., Grebogi, C., Kurths, J.: Modeling of deterministic chaotic systems. Phys. Rev. E 59, 2907 (1999)ADSCrossRefGoogle Scholar
  20. 20.
    Mancho, A.M., Wiggins, S., Curbelo, J., Mendoza, C.: Lagrangian descriptors: a method for revealing phase space structures of general time dependent dynamical systems. Commun. Nonlinear Sci. 18, 3530 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Meiss, J.D.: Transient measures for the standard map. Phys. D 74, 254 (1994)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Oyarzabal, R.S., Szezech, J.D., Batista, A.M., de Souza, S.L.T., Caldas, I.L., Viana, R.L., Sanjuán, M.A.F.: Transient chaotic transport in dissipative drift motion. Phys. Lett. A 380, 1621 (2016)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Parisi, G., Vulpiani, A.: Scaling law for the maximal Lyapunov characteristic exponent of infinite product of random matrices. J. Phys. A 19, L45 (1986)CrossRefGoogle Scholar
  24. 24.
    Prasad, A., Ramaswany, R.: Characteristic distributions of finite-time Lyapunov exponents. Phys. Rev. E 60, 2761 (1999)ADSCrossRefGoogle Scholar
  25. 25.
    Saiki, Y., Sanjuán, M.A.F.: Low-dimensional paradigms for high-dimensional hetero-chaos. Chaos 28, 103110 (2018)ADSMathSciNetCrossRefGoogle 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.: Chaotic itinerancy based on attractors of one-dimensional maps. Chaos 13, 947 (2003)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Sauer, T., Grebogi, C., Yorke, J.A.: How long do numerical chaotic solutions remain valid? Phys. Lett. A 79, 59 (1997)CrossRefGoogle Scholar
  29. 29.
    Skokos, Ch., Bountis, T.C., Antonopoulos Ch.: Geometrical properties of local dynamics in Hamiltonian systems: the generalized alignment index (GALI) method. Phys. D 231, 30 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Smith, L.A., Spiegel, E.A.: Strange accumulators. In: Buchler, J.R., Eichhorn, H. (eds.) Chaotic Phenomena in Astrophysics. New York Academy of Sciences, New York (1987)Google Scholar
  31. 31.
    Stefanski, K., Buszko, K., Piecsyk, K.: Transient chaos measurements using finite-time Lyapunov Exponents. Chaos 20, 033117 (2010)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Szezech, Jr.J.D., Lopes, S.R., Viana, R.L.: Finite time Lyapunov spectrum for chaotic orbits of non integrable hamiltonian systems. Phys. Lett. A 335, 394 (2005)Google Scholar
  33. 33.
    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
  34. 34.
    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
  35. 35.
    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
  36. 36.
    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
  37. 37.
    Viana, R.L., Grebogi, C.: Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462 (2000)ADSCrossRefGoogle Scholar
  38. 38.
    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
  39. 39.
    Viana, R.L., Barbosa, J.R., Grebogi, C., Batista, C.M.: Simulating a chaotic process. Braz. J. Phys. 35, 1 (2005)ADSCrossRefGoogle Scholar
  40. 40.
    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
  41. 41.
    Ziehmann, C., Smith, L.A., Kurths, J.: Localized Lyapunov exponents and the prediction of predictability. Phys. Lett. A 271, 237 (2000)ADSMathSciNetCrossRefGoogle 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