Skip to main content

Lyapunov Exponents

  • Chapter
  • First Online:
  • 737 Accesses

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

Abstract

We have seen in the previous chapter that one of the fundamental questions about the dynamics of a system is to know whether it is predictable or not. The answer to this question is tightly related to analyse if chaos is present in the dynamical flow.

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. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems. Springer, New-York (1996)

    Google Scholar 

  2. Anteneodo, C.: Statistics of finite-time Lyapunov exponents in the Ulam map. Phys. Rev. E 69, 016207 (2004)

    Article  ADS  Google Scholar 

  3. Araujo, T., Mendes, R.V., Seixas, J.: A dynamical characterization of the small world phase. Phys. Lett. A 319, 285 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Aurell, E., Boffeta, G., Crisanti, A., Paladin, G., Vulpiani, A.: Predictability in the large: an extension of the concept of Lyapunov exponent. J. Phys. A Math. Gen. 30, 1 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Badii, R., Heinzelmann, K., Meier, P.F., Politi, A.: Correlation functions and generalized Lyapunov exponents. Phys. Rev. A 37, 1323 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Meccanica 9, 20 (1980)

    MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. Boffetta, G., Cencini, M., Falcioni, M., Vulpiani, A.: Predictability: a way to characterize complexity. Phys. Rep. 356, 367 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Buizza, R., Palmer, T.: The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci. 52, 1434 (1995)

    Article  ADS  Google Scholar 

  11. Carpintero, D.D., Aguilar, L.A.: Orbit classification in arbitrary 2D and 3D potentials. Mon. Not. R. Astron. Soc. 298, 21 (1998)

    Article  ADS  Google Scholar 

  12. Cvitanovic, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G., Whelan, N., Wirzba, A.: Chaos: Classical and Quantum. ChaosBook.org, Niels Bohr Institute, Copenhagen (2016)

    Google Scholar 

  13. Contopoulos, G., Voglis, N.: A fast method for distinguishing between ordered and chaotic orbits. Astron. Astrophys. 317, 317 (1997)

    Google Scholar 

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

    ADS  Google Scholar 

  15. Crisanti, A., Paladin, G., Vulpiani, A.: Product of Random Matrices. Springer Series in Solid State Sciences. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  16. Custodio, M.S., Manchein, C., Beims, M.W.: Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems. Chaos 22, 026112 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Diakonos, F.K., Pingel, D., Schmelcher, P.: Analyzing Lyapunov spectra of chaotic dynamical systems. Phys. Rev. E 62, 4413 (2000)

    Article  ADS  Google Scholar 

  18. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Ershov, S.V., Potapov, A.B.: On the nature of nonchaotic turbulence. Phys. Lett. A 167, 60 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  20. Ershov, S.V., Potapov, A.B.: On the concept of stationary Lyapunov basis. Phys. D 118, 167 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Finn, J.M., Hanson, J.D., Kan, I., Ott, E.: Steady fast dynamo flows. Phys. Fluids B 3, 1250 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  22. Froeschlé, C., Lohinger, E.: Generalized Lyappunov characteristic indicators and corresponding Kolmogorov like entrophy of the standard mapping. Celest. Mech. Dyn. Astron. 56, 307 (1993)

    Article  ADS  Google Scholar 

  23. Froyland, G., Huls, T., Morriss, G.P., Watson, T.M.: Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: a comparative numerical study. Phys. D 247, 18–39 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fujisaka, H.: Statistical dynamics generated by fluctuations of local Lyapunov exponents. Prog. Theor. Phys. 70, 1264 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Gao, J.B., Hu, J., Tung, W.W., Cao, Y.H.: Distinguishing chaos from noise by scale-dependent Lyapunov exponents. Phys. Rev. E 74, 066204 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  26. Ginelli, F., Poggi, P., Turchi, A., Chate, H., Livi, R., Politi, A.: Characterizing dynamics with covariant Lyapunov vectors. Phys. Rev. Lett. 99, 130601 (2007)

    Article  ADS  Google Scholar 

  27. 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  MATH  Google Scholar 

  28. Haller, G.: Distinguished material surfaces and coherent structures in 3D fluid flows. Phys. D 149, 248 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Heggie, D.C.: Chaos in the N-body problem of stellar dynamics. In: Roy, A.E. (ed.) Predictability, Stability and Chaos in N-Body Dynamical Systems. Plenum Press, New York (1991)

    Google Scholar 

  30. Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  31. Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  32. Kalnay, E., Corazza, M., Cai, M.: Are bred vectors the same as Lyapunov vectors? EGS XXVII General Assembly, Nice, 21–26 Apr 2002, abstract 6820

    Google Scholar 

  33. Kandrup, H.E., Mahon, M.E.: Short times characterisations of stochasticity in nonintegrable galactic potentials. Astron. Astrophys. 290, 762 (1994)

    ADS  Google Scholar 

  34. Kapitakinak, T.: Generating strange nonchaotic trajectories. Phys. Rev. E 47, 1408 (1993)

    Article  ADS  Google Scholar 

  35. Kaplan, J.L., Yorke, J.A.: Chaotic behavior of multidimensional difference equations. In: Peitgen, H.O., Walter, H.O. (eds.) Functional Differential Equations and Approximations of Fixed Points. Lecture Notes in Mathematics, vol. 730. Springer, Berlin (1979)

    Google Scholar 

  36. Klages, R.: Weak chaos, infinite ergodic theory, and anomalous dynamics. In: Leoncini, X., Leonetti, M. (eds.) From Hamiltonian Chaos to Complex Systems. Springer, Berlin (2013)

    MATH  Google Scholar 

  37. Klein, M., Baier, G.: Hierarchies of dynamical systems. In: Baier, G., Klein, M. (eds.) A Chaotic Hierarchy. World Scientific, Singapore (1991)

    MATH  Google Scholar 

  38. Kocarev, L., Szcepanski, J.: Finite-space Lyapunov exponents and pseudochaos. Phys. Rev. Lett. 93, 234101 (2004)

    Article  ADS  Google Scholar 

  39. Kostelich, E.J., Kan, I., Grebogi, C., Ott, E., Yorke, J.A.: Unstable dimension variability: a source of nonhyperbolicity in chaotic systems. Phys. D 109, 81 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kuptsov, P.V., Parlitz, U.: Theory and computation of covariant Lyapunov vectors. J. Nonlin. Sci. 22, 727 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  42. Legras, B., Vautard, R.: A guide to Lyapunov vectors. In: Palmer, T. (ed.) Predictability Seminar Proceedings, ECWF Seminar, vol. 1, pp. 135–146. European Centre for Medium-Range Weather Forecasts, Reading (1996)

    Google Scholar 

  43. Lyapunov, A.M.: The General Problem of the Stability of Motion. Taylor and Francis, London (1992) (English translation from the French 1907, in turn from the Russian 1892)

    Book  MATH  Google Scholar 

  44. Lepri, S., Politi, A., Torcini, A.: Chronotropic Lyapunov analysis: (I) a comprehensive characterization of 1D systems. J. Stat. Phys. 82, 1429 (1996)

    Article  ADS  MATH  Google Scholar 

  45. Mahon, M.E., Abernathy, R.A., Bradley, B.O., Kandrup, H.E.: Transient ensemble dynamics in time-independent galactic potentials. Mon. Not. R. Astron. Soc. 275, 443 (1995)

    Article  ADS  Google Scholar 

  46. Mitchell, L., Gottwald, G.A.: On finite size Lyapunov exponents in multiscale systems. Chaos 22, 23115 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  47. Mosekilde, E.: Topics in nonlinear dynamics: applications to physics, biology and economic. World Scientific Publishing, Singapore (1996)

    MATH  Google Scholar 

  48. Moser, H.R., Meier, P.F.: The structure of a Lyapunov spectrum can be determined locally. Phys. Let. A 263, 167 (1999)

    Article  ADS  Google Scholar 

  49. Motter, A.E.: Relativistic chaos is coordinate invariant. Phys. Rev. Lett. 91, 23 (2003)

    Article  Google Scholar 

  50. Mulansky, M., Ahnert, K., Pikovsky, A., Shepelyansky, D.L.: Strong and weak chaos in weakly nonintegrable many-body Hamiltonian systems. J. Stat. Phys. 145, 1256 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Okushima, T.: New method for computing finite-time Lyapuunov exponents. Phys. Rev. Lett. 91, 25 (2003)

    Article  Google Scholar 

  52. Oseledec, V.I.: A multiplicative ergodic theorem. Moscow Math. Soc. 19, 197 (1968)

    MathSciNet  Google Scholar 

  53. Ott, E.: Chaos in Dynamical Systems, 2nd ed. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  54. Ott, W., Yorke, J.A.: When Lyapunov exponents fail to exist. Phys. Rev. E 78, 056203 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  55. 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 

  56. Patsis, P.A., Efthymiopoulos, C., Contopoulos, G., Voglis, N.: Dynamical spectra of barred galaxies. Astron. Astrophys. 326, 493 (1997)

    ADS  Google Scholar 

  57. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  58. Pesin, Y.: Dimension Theory in Dynamical Systems, Rigourous Results and Applications. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

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

    Article  ADS  Google Scholar 

  60. Prasad, A., Ramaswamy, R.: Finite-time Lyapunov exponents of strange nonchaotic attractors. In: Daniel, M., Tamizhmani, K., Sahadevan, R. (eds.) Nonlinear Dynamics: Integrability and Chaos. Narosa, New Delhi (2000)

    Google Scholar 

  61. Ramaswamy, R.: Symmetry breaking in local Lyapunov exponents. Eur. Phys. J.B. 29, 339 (2002)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. Sandri, M.: Numerical calculation of Lyapunov exponents. Math. J. 6(3), 78–84 (1996)

    Google Scholar 

  64. Siopis, C., Kandrup, H.E., Contopoulos, G., Dvorak, R.: Universal properties of escape in dynamical systems. Celest. Mech. Dyn. Astron. 65, 57 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  65. Smith, H., Contopoulos, G.: Spectra of stretching numbers of orbits in oscillating galaxies. Astron. Astrophys. 314, 795 (1996)

    ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  67. Toth, Z., Kalnay, E.: Ensemble forecasting at NCEP and the breeding method. Mon. Weather Rev. 125, 3297 (1997)

    Article  ADS  Google Scholar 

  68. Tsiganis, K., Anastasiadis, A., Varvoglis, H.: Dimensionality differences between sticky and non-sticky chaotic trajectory segments in a 3D Hamiltonian system. Chaos Solitons Fractals 11, 2281 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. Vallejo, J.C., Aguirre, J., Sanjuán, M.A.F.: Characterization of the local instability in the Henon-Heiles Hamiltonian. Phys. Lett. A 311, 26 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. Vallejo, J.C., Viana, R., Sanjuán, 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 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  73. Voglis, N., Contopoulos, G.: Invariant spectra of orbits in dynamical systems. J. Phys. A27, 4899 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  74. Voglis, N., Contopoulos, G., Efthymioupoulos, C.: Method for distinguishing between ordered and chaotic orbits in four-dimensional maps. Phys. Rev. E 57, 372 (1998)

    Article  ADS  Google Scholar 

  75. Vozikis, C., Varvoglis, H., Tsiganis, K.: The power spectrum of geodesic divergences as an early detector of chaotic motion. Astron. Astrophys. 359, 386 (2000)

    ADS  Google Scholar 

  76. Weisstein, E.W.: Lyapunov characteristic exponent. From MathWorld A Wolfram Web resource. http://mathworld.wolfram.com/LyapunovCharacteristicExponent.html

  77. Wolfe, C.L., Samelson, R.M.: An efficient method for recovering Lyapunov vectors from singular vectors. Tellus A 59A, 355 (2007)

    Article  ADS  Google Scholar 

  78. Xu, M., Paul, M.R.: Covariant Lyapunov vectors of chaotic Rayleigh-Benard convection. Phys. Rev. E 93, 062208 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  79. Yanchuk, S., Kapitaniak, T.: Chaos-hyperchaos transition in coupled Rössler systems. Phys. Lett. A 290, 139 (2001)

    Article  ADS  MATH  Google Scholar 

  80. 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 

  81. Yang, H.: Dependence of Hamiltonian chaos on perturbation structure. Int. J. Bifurcation Chaos 3, 1013 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  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). Lyapunov Exponents. In: Predictability of Chaotic Dynamics . Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-030-28630-9_2

Download citation

Publish with us

Policies and ethics