Skip to main content

Dynamical Regimes and Timescales

  • Chapter
  • First Online:
  • 669 Accesses

Part of the book series: Springer Series in Synergetics ((SSSYN))

Abstract

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

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Abraham, R., Smale S.: Non-genericity of Ω-stability. Proc. Symp. Pure Math. 14, 5 (1970)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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. Alligood, K.T., Sander, E., Yorke, J.A.: Three-dimensional crisis: crossing bifurcations and unstable dimension variability. Phys. Rev. Lett. 96, 244103 (2006)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  6. Benzi, R., Parisi, G., Vulpiani, A.: Characterisation of intermittency in chaotic systems. J. Phys. A, 18, 2157 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  7. Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press, Princeton (1987)

    MATH  Google Scholar 

  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. Contopoulos, G.: Orbits in highly perturbed dynamical systems. I. Periodic orbits. Astron. J. 75, 96 (1970)

    ADS  MathSciNet  Google Scholar 

  10. Contopoulos, G., Grousousakou, E., Voglis, N.: Invariant spectra in hamiltonian systems. Astron. Astrophys. 304, 374 (1995)

    ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  13. Grassberger, P.: Generalizations of the Hausdorff dimension of fractal measures. Phys. Lett. A 107, 101 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  14. Grassberger, P., Badii, R., Politi, A.: Scaling Laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51, 135 (1988)

    Article  ADS  Google Scholar 

  15. Grebogi, C., Ott, E., Yorke, J.A.: Crises, sudden changes in chaotic attractors, and transient chaos. Phys. D 7, 181 (1983)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  19. Lai, Y.C., Grebogi, C., Kurths, J.: Modeling of deterministic chaotic systems. Phys. Rev. E 59, 2907 (1999)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  21. Meiss, J.D.: Transient measures for the standard map. Phys. D 74, 254 (1994)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  24. Prasad, A., Ramaswany, R.: Characteristic distributions of finite-time Lyapunov exponents. Phys. Rev. E 60, 2761 (1999)

    Article  ADS  Google Scholar 

  25. Saiki, Y., Sanjuán, M.A.F.: Low-dimensional paradigms for high-dimensional hetero-chaos. Chaos 28, 103110 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  26. Sauer, T.: Shadowing breakdown and large errors in dynamical simulations of physical systems. Phys. Rev. E. 65, 036220 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  27. Sauer, T.: Chaotic itinerancy based on attractors of one-dimensional maps. Chaos 13, 947 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  28. Sauer, T., Grebogi, C., Yorke, J.A.: How long do numerical chaotic solutions remain valid? Phys. Lett. A 79, 59 (1997)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Stefanski, K., Buszko, K., Piecsyk, K.: Transient chaos measurements using finite-time Lyapunov Exponents. Chaos 20, 033117 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  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. Vallejo, J.C., Sanjuan, M.A.F.: Predictability of orbits in coupled systems through finite-time Lyapunov exponents. New J. Phys. 15, 113064 (2013)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  37. Viana, R.L., Grebogi, C.: Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462 (2000)

    Article  ADS  Google Scholar 

  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. Viana, R.L., Barbosa, J.R., Grebogi, C., Batista, C.M.: Simulating a chaotic process. Braz. J. Phys. 35, 1 (2005)

    Article  ADS  Google Scholar 

  40. Yanchuk, S., Kapitaniak, T.: Symmetry increasing bifurcation as a predictor of chaos-hyperchaos transition in coupled systems. Phys. Rev. E 64 056235 (2001)

    Article  ADS  Google Scholar 

  41. Ziehmann, C., Smith, L.A., Kurths, J.: Localized Lyapunov exponents and the prediction of predictability. Phys. Lett. A 271, 237 (2000)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Vallejo, J.C., Sanjuan, M.A.F. (2019). Dynamical Regimes and Timescales. In: Predictability of Chaotic Dynamics . Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-030-28630-9_3

Download citation

Publish with us

Policies and ethics