Advertisement

Dynamical Regimes and Time Scales

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

Abstract

The key factor to build the finite-time distributions is finding the most adequate interval length, to be large enough to ensure a satisfactory reduction of the local fluctuations, but small enough to reveal slow trends. This length is different for every orbit. There are different time scales to take into account in every model, including transient behaviours that could be of interest. By a proper selection of the total integration time, we can characterise the dynamics using small finite-time interval lengths. But increasing those lengths, we see how the distributions stop tracing the flow at local scales and begin to describe the flow at global scales, that is, the global regime.

Keywords

Lyapunov Exponent Unstable Manifold Deviation Vector Chaotic Orbit Unstable Periodic 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.

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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Binney, J., Tremaine, S.: Galactic Dynamics. Princenton University Press, Princenton (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 Proc. Geophys. 17, 1–36 (2010)ADSCrossRefGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grassberger, P.: Generalizations of the Hausdorff dimension of fractal measures. Phys. Lett. A 107, 101 (1985)ADSMathSciNetCrossRefzbMATHGoogle 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.
    Grebogi, C., Ott, E., Yorke, J.A.: Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 7, 181 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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
  16. 16.
    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
  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.
    Lai, Y.C., Grebogi, C., Kurths, J.: Modeling of deterministic chaotic systems. Phys. Rev. E 59, 2907 (1999)ADSCrossRefGoogle Scholar
  19. 19.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Meiss, J.D.: Transient measures for the standard map. Physica D 74, 254 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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
  22. 22.
    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
  23. 23.
    Prasad, A., Ramaswany, R.: Characteristic distributions of finite-time Lyapunov exponents. Phys. Rev. E 60, 2761 (1999)ADSCrossRefGoogle Scholar
  24. 24.
    Sauer, T.: Shadowing breakdown and large errors in dynamical simulations of physical systems. Phys. Rev. E 65, 036220 (2002)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Sauer, T.: Chaotic itinerancy based on attractors of one-dimensional maps. Chaos 13, 947 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sauer, T., Grebogi, C., Yorke, J.A.: How long do numerical chaotic solutions remain valid? Phys. Lett. A 79, 59 (1997)CrossRefGoogle Scholar
  27. 27.
    Skokos, Ch., Bountis, T.C., Antonopoulos, Ch.: Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method. Physica D 231, 30 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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
  29. 29.
    Stefanski, K., Buszko, K., Piecsyk, K.: Transient chaos measurements using finite-time Lyapunov exponents. Chaos 20, 033117 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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)ADSMathSciNetCrossRefzbMATHGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    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
  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., Pinto, S.E., Barbosa, J.R., Grebogi, C.: Pseudo-deterministic chaotic systems. Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, 1 (2003)Google Scholar
  35. 35.
    Viana, R.L., Barbosa, J.R., Grebogi, C., Batista, C.M.: Simulating a chaotic process. Braz. J. Phys. 35, 1 (2005)ADSCrossRefGoogle Scholar
  36. 36.
    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
  37. 37.
    Ziehmann, C., Smith, L.A., Kurths, J.: Localized Lyapunov exponents and the prediction of predictability. Phys. Lett. A 271, 237 (2000)ADSMathSciNetCrossRefzbMATHGoogle 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

Personalised recommendations