Lyapunov Exponents

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


Lyapunov exponents are a well-known diagnostic tool for analysing the presence of chaos in a system. They provide a quantitative measure of the divergence or convergence of nearby trajectories, by averaging the expansion rate of the phase space. In practice, the calculation is performed numerically and a finite-time integration time is used instead of the necessary infinite time. Moreover, the convergence of the averaging process towards the final asymptotic value can be very long. The distributions of finite-time Lyapunov exponents exploit the dependence on the finite-time for tracing the dynamics of the flow, and by analysing how their shapes change, we get insight into the specific nature of the orbit.


Lyapunov Exponent Interval Size Lyapunov Spectrum Distortion Tensor Lyapunov Vector 
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.
    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
  2. 2.
    Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems, p. 383. Springer, New York (1996)Google Scholar
  3. 3.
    Anteneodo, C.: Statistics of finite-time Lyapunov exponents in the Ulam map. Phys. Rev. E 69, 016207 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    Araujo, T., Mendes, R.V., Seixas, J.: A dynamical characterization of the small world phase. Phys. Lett. A 319, 285 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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), 1–26 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Badii, R., Heinzelmann, K., Meier, P.F., Politi, A.: Correlation functions and generalized Lyapunov exponents. Phys. Rev. A 37, 1323 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)zbMATHGoogle Scholar
  8. 8.
    Benzi, R., Parisi, G., Vulpiani, A.: Characterisation of intermittency in chaotic systems. J. Phys. A 18, 2157 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press, Princeton, NJ (1987)zbMATHGoogle Scholar
  10. 10.
    Boffetta, G., Cencini, M., Falcioni, M., Vulpiani, A.: Predictability: a way to characterize complexity. Phys. Rep. 356, 367 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carpintero, D.D., Aguilar, L.A.: Orbit classification in arbitrary 2D and 3D potentials. Mon. Not. R. Astron. Soc. 298, 21 (1998)ADSCrossRefGoogle Scholar
  12. 12.
    Contopoulos, G., Voglis, N.: A fast method for distinguishing between ordered and chaotic orbits. Astron. Astrophys. 317, 317 (1997)Google Scholar
  13. 13.
    Contopoulos, G., Grousousakou, E., Voglis, N.: Invariant spectra in Hamiltonian systems. Astron. Astrophys. 304, 374 (1995)ADSGoogle Scholar
  14. 14.
    Crisanti, A., Paladin, G., Vulpiani, A.: Product of Random Matrices. Springer Series in Solid State Sciences. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  15. 15.
    Custodio, M.S., Manchein, C., Beims, M.W.: Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems. Chaos 22, 026112 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cvitanovic, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G., Whelan, N., Wirzba, A.: Chaos: Classical and Quantum. Niels Bohr Institute, Copenhagen (2016). ChaosBook.orgGoogle Scholar
  17. 17.
    Diakonos, F.K., Pingel, D., Schmelcher, P.: analysing Lyapunov spectra of chaotic dynamical systems. Phys. Rev. E 62, 4413 (2000)Google Scholar
  18. 18.
    Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ershov, S.V., Potapov, A.B.: On the nature of nonchaotic turbulence. Phys. Lett. A 167, 60 (1992)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Ershov, S.V., Potapov, A.B.: On the concept of stationary Lyapunov basis. Physica D 118, 167 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Finn, J.M., Hanson, J.D., Kan, I., Ott, E.: Steady fast dynamo flows. Phys. Fluids B 3, 1250 (1991)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Froeschlé, C., Lohinger, E.: Generalized Lyapunov characteristic indicators and corresponding Kolmogorov like entropy of the standard mapping. Celest. Mech. Dyn. Astron. 56, 307 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fujisaka, H.: Statistical dynamics generated by fluctuations of local Lyapunov exponents. Prog. Theor. Phys. 70, 1264 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    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
  26. 26.
    Haller, G.: Distinguished material surfaces and coherent structures in 3d fluid flows. Physica D 149, 248 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73 (1964)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, Cambridge (2007)Google Scholar
  29. 29.
    Kalnay, E., Corazza, M., Cai, M.: Are bred vectors the same as Lyapunov vectors? EGS XXVII General Assembly, Nice, 21–26 April 2002. Abstract 6820Google Scholar
  30. 30.
    Kandrup, H.E., Mahon, M.E.: Short times characterisations of stochasticity in nonintegrable galactic potentials. Astron. Astrophys. 290, 762 (1994)ADSGoogle Scholar
  31. 31.
    Kapitakinak, T.: Generating strange nonchaotic trajectories. Phys. Rev. E 47, 1408 (1993)ADSCrossRefGoogle Scholar
  32. 32.
    Kaplan, J.L., Yorke, J.A.: Chaotic behaviour 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, p. 204. Springer, Berlin (1979)Google Scholar
  33. 33.
    Klages, R.: Weak chaos, infinite ergodic theory, and anomalous dynamics. In: Leoncini, X., Leonetti, M. (eds.) From Hamiltonian Chaos to Complex Systems, pp. 3–42. Springer, Berlin (2013). ISBN 978-1-4614-6961-2CrossRefGoogle Scholar
  34. 34.
    Klein, M., Baier, G.: Hierarchies of dynamical systems. In: Baier, G., Klein, M. (eds.) A Chaotic Hierarchy. World Scientific, Singapore (1991)Google Scholar
  35. 35.
    Kocarev, L., Szcepanski, J.: Finite-space Lyapunov exponents and pseudoChaos. Phys. Rev. Lett. 93, 234101 (2004)ADSCrossRefGoogle Scholar
  36. 36.
    Kostelich, E.J., Kan, I., Grebogi, C., Ott, E., Yorke, J.A.: Unstable dimension variability: a source of nonhyperbolicity in chaotic systems. Physica D 109, 81 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Lai, Y.C., Grebogi, C., Kurths, J.: Modeling of deterministic chaotic systems. Phys. Rev. E 59, 2907 (1999)ADSCrossRefGoogle Scholar
  38. 38.
    Lepri, S., Politi, A., Torcini, A.: Chronotropic Lyapunov analysis: (I) a comprehensive characterization of 1D systems. J. Stat. Phys. 82, 1429 (1996)ADSCrossRefzbMATHGoogle Scholar
  39. 39.
    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 1892Google Scholar
  40. 40.
    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)ADSGoogle Scholar
  41. 41.
    Mitchell, L., Gottwald, G.A.: On finite size Lyapunov exponents in multiscale systems. Chaos 22, 23115 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Mosekilde, E.: Topics in Nonlinear Dynamics: Applications to Physics, Biology and Economic. World Scientific, Singapore (1996)zbMATHGoogle Scholar
  43. 43.
    Moser, H.R., Meier, P.F.: The structure of a Lyapunov spectrum can be determined locally. Phys. Lett. A 263, 167 (1999)ADSCrossRefGoogle Scholar
  44. 44.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Okushima, T.: New method for computing finite-time Lyapunov exponents. Phys. Rev. Lett. 91, 25 (2003)CrossRefGoogle Scholar
  46. 46.
    Oseledec, V.I.: A multiplicative ergodic theorem. Mosc. Math. Soc. 19, 197 (1968)MathSciNetGoogle Scholar
  47. 47.
    Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  48. 48.
    Ott, W., Yorke, J.A.: When Lyapunov exponents fail to exist. Phys. Rev. E 78, 056203 (2008)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    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
  50. 50.
    Patsis, P.A., Efthymiopoulos, C., Contopoulos, G., Voglis, N.: Dynamical spectra of barred galaxies. Astron. Astrophys. 326, 493 (1997)ADSGoogle Scholar
  51. 51.
    Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Pesin, Y.: Dimension Theory in Dynamical Systems. Rigorous Results and Applications. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  53. 53.
    Prasad, A., Ramaswany, R.: Characteristic distributions of finite-time Lyapunov exponents. Phys. Rev. E 60, 2761 (1999)ADSCrossRefGoogle Scholar
  54. 54.
    Prasad, A., Ramaswamy, R.: Finite-time Lyapunov exponents of strange nonchaotic attractors. In: Eds. Daniel, M., Tamizhmani, K., Sahadevan, R. (eds.) Nonlinear Dynamics: Integrability and Chaos, pp. 227–234. Narosa, New Delhi (2000)Google Scholar
  55. 55.
    Ramaswamy, R.: Symmetry breaking in local Lyapunov exponents. Eur. Phys. J. B. 29, 339 (2002)ADSCrossRefGoogle Scholar
  56. 56.
    Sandri, M.: Numerical calculation of Lyapunov exponents. Math. J. 6 (3), 78–84 (1996)Google Scholar
  57. 57.
    Siopis, C., Kandrup, H.E., Contopoulos, G., Dvorak, R.: Universal properties of escape in dynamical systems. Celest. Mech. Dyn. Astron. 65, 57 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Smith, H., Contopoulos, G.: Spectra of stretching numbers of orbits in oscillating galaxies. Astron. Astrophys. 314, 795 (1996)ADSGoogle Scholar
  59. 59.
    Stefanski, K., Buszko, K., Piecsyk, K.: Transient chaos measurements using finite-time Lyapunov Exponents. Chaos 20, 033117 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Tsiganis, K., Anastasiadis, A., Varvoglis, H.: Dimensionality differences between sticky and non-sticky chaotic trajectory segments in a 3D Hamiltonian system. Chaos, Solitons and Fractals 11, 2281–2292 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    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
  62. 62.
    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
  63. 63.
    Viana, R.L., Grebogi, C.: Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462 (2000)ADSCrossRefGoogle Scholar
  64. 64.
    Voglis, N., Contopoulos, G.: Invariant spectra of orbits in dynamical systems. J. Phys. A27, 4899 (1994)ADSMathSciNetzbMATHGoogle Scholar
  65. 65.
    Voglis, N., Contopoulos, G., Efthymioupoulos, C.: Method for distinguishing between ordered and chaotic orbits in four-dimensional maps. Phys. Rev. E 57, 372 (1998)ADSCrossRefGoogle Scholar
  66. 66.
    Vozikis, Ch., Varvoglis, H., Tsiganis, K.: The power spectrum of geodesic divergences as an early detector of chaotic motion. Astron. Astrophys. 359, 386 (2000)ADSGoogle Scholar
  67. 67.
    Weisstein, E.W.: Lyapunov characteristic exponent, from MathWorld A Wolfram Web resource (2015). Google Scholar
  68. 68.
    Yanchuk, S., Kapitaniak, T.: Chaos-hyperchaos transition in coupled Rössler systems. Phys. Lett. A 290, 139 (2001)ADSCrossRefzbMATHGoogle Scholar
  69. 69.
    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
  70. 70.
    Yang, H.: Dependence of Hamiltonian Chaos on perturbation structure. Int. J. Bifurcation Chaos 3, 1013 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Ziehmann, C., Smith, L.A., Kurths, J.: Localized Lyapunov exponents and the prediction of predictability. Phys. Lett. A 271, 237 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© 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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