The Lyapunov Characteristic Exponents and Their Computation

Part of the Lecture Notes in Physics book series (LNP, volume 790)

Abstract

We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec [102], which provides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been extensively used as an indicator of chaos, and the algorithm of the so-called standard method, developed by Benettin et al. [14], for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although we are mainly interested in finite-dimensional conservative systems, i.e., autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allen, L., Bridges, T.J.: Numerical exterior algebra and the compound matrix method. Numerische Mathematik 92, 197–232 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Antonopoulos, C., Bountis, T.: Detecting order and chaos by the linear dependence index (LDI) method. ROMAI J. 2, 1–13 (2006)MathSciNetGoogle Scholar
  3. 3.
    Antonopoulos, C., Bountis, T., Skokos, Ch. Chaotic dynamics of N-degree of freedom Hamiltonian systems. Int. J. Bif. Chaos 16, 1777–1793 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bario, R.: Sensitivity tools vs. Poincaré sections. Chaos Solit. Fract. 25, 711–726 (2005)CrossRefADSGoogle Scholar
  5. 5.
    Bario, R.: Painting chaos: a gallery of sensitivity plots of classical problems. Int. J. Bif. Chaos 16, 2777–2798 (2006)CrossRefGoogle Scholar
  6. 6.
    Bario, R., Borczyk, W., Breiter, S.: Spurious structures in chaos indicators maps. Chaos Solit. Fract. (in press) (2009)Google Scholar
  7. 7.
    Barreira, L., Pesin, Y.: Smooth ergodic theory and nonuniformly hyperbolic dynamics. In: Hasselblatt, B., Katok, A. (eds.): Handbook of Dynamical Systems, vol. 1B, 57–263. Elsevier (2006)Google Scholar
  8. 8.
    Benettin, G., Strelcyn, J.-M.: Numerical experiments of the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy. Phys. Rev. A 17, 773–785 (1978)ADSCrossRefGoogle Scholar
  9. 9.
    Benettin, G., Galgani, L.: Lyapunov characteristic exponents and stochasticity. In: Laval, G., Grésillon, D. (eds.): Intrinsic Stochasticity in Plasmas, 93–114, Edit. Phys. Orsay (1979)Google Scholar
  10. 10.
    Benettin, G., Galgani, L., Strelcyn, J.-M.: Kolmogorov entropy and numerical experiments. Phys. Rev. A 14, 2338–2344 (1976)ADSCrossRefGoogle Scholar
  11. 11.
    Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M.: Tous les nombres caractéristiques sont effectivement calculables. C. R. Acad. Sc. Paris Sér. A 286, 431–433 (1978)MathSciNetADSMATHGoogle Scholar
  12. 12.
    Benettin, G., Froeschlé, C., Scheidecker, J.P.: Kolmogorov entropy of a dynamical system with an increasing number of degrees of freedom. Phys. Rev. A 19, 2454–2460 (1979)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    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. Part 1: theory. Meccanica March: 9–20 (1980)Google Scholar
  14. 14.
    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. Part 2: Numerical application. Meccanica March: 21–30 (1980)Google Scholar
  15. 15.
    Bountis, T., Manos, T., Christodoulidi, H.: Application of the GALI method to localization dynamics in nonlinear systems. J. Comp. Appl. Math. 227, 17–26 (2009), nlin.CD/0806.3563 (2008)Google Scholar
  16. 16.
    Bourbaki, N.: Éléments de mathématique, Livre II: Algèbre, Chapitre 3. Hermann, Paris (1958)Google Scholar
  17. 17.
    Brown, R., Bryant, P., Abarbanel, H.D.I.: Computing the Lyapunov spectrum of a dynamical system from an observed time series. Phys. Rev. A 43, 2787–2806 (1991)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Bridges, T.J., Reich, S.: Computing Lyapunov exponents on a Stiefel manifold. Physica D 156, 219–238 (2001)MathSciNetADSCrossRefMATHGoogle Scholar
  19. 19.
    Burns, K., Donnay, V.: Embedded surfaces with ergodic geodesic flow. Int. J. Bif. Chaos 7, 1509–1527 (1997)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Carbonell, F., Jimenez, J.C., Biscay, R.: A numerical method for the computation of the Lyapunov exponents of nonlinear ordinary differential equations. Appl. Math. Comput. 131, 21–37 (2002)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Casartelli, M., Diana, E., Galgani, L., Scotti, A.: Numerical computations on a stochastic parameter related to the Kolmogorov entropy. Phys. Rev. A 13, 1921–1925 (1976)ADSCrossRefGoogle Scholar
  22. 22.
    Casati, G., Ford, J.: Stochastic transition in the unequal-mass Toda lattice. Phys. Rev. A 12, 1702–1709 (1975)ADSCrossRefGoogle Scholar
  23. 23.
    Casati, G., Chirikov, B.V., Ford, J.: Marginal local instability of quasi-periodic motion. Phys. Let. A 77, 91–94 (1980)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Chen, Z.–M., Djidjeli, K., Price, W.G.: Computing Lyapunov exponents based on the solution expression of the variational system. Appl. Math. Comput. 174, 982–996 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Chernov, N., Markarian, R.: Chaotic billiards. Mathematical Surveys and Monographs, Vol. 127. American Mathematical Society (2006)Google Scholar
  26. 26.
    Christiansen, F., Rugh, H.H.: Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization. Nonlinearity 10, 1063–1072 (1997)MathSciNetADSCrossRefMATHGoogle Scholar
  27. 27.
    Christodoulidi, H., Bountis, T.: Low-dimensional quasiperiodic motion in Hamiltonian systems. ROMAI J. 2, 37–44 (2006)MathSciNetGoogle Scholar
  28. 28.
    Cincotta, P.M., Simó, C.: Simple tools to study global dynamics in non-axisymmetric galactic potentials-I. Astron. Astrophs. Supp. Ser. 147, 205–228 (2000)ADSCrossRefGoogle Scholar
  29. 29.
    Cincotta, P.M., Giordano, C.M., Simó, C.: Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits. Physica D 182, 151–178 (2003)MathSciNetADSCrossRefMATHGoogle Scholar
  30. 30.
    Contopoulos, G.: Order and Chaos in Dynamical Astronomy. Springer, Berlin Heidelberg New York (2002)MATHGoogle Scholar
  31. 31.
    Contopoulos, G., Giorgilli, A.: Bifurcations and complex instability in a 4-dimensional symplectic mapping. Meccanica 23, 19–28 (1988)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Contopoulos, G., Voglis, N.: Spectra of stretching numbers and helicity angles in dynamical systems. Cel. Mech. Dyn. Astron. 64, 1–20 (1996)MathSciNetADSCrossRefMATHGoogle Scholar
  33. 33.
    Contopoulos, G., Voglis, N.: A fast method for distinguishing between ordered and chaotic orbits. Astron. Astrophs. 317, 73–81 (1997)ADSGoogle Scholar
  34. 34.
    Contopoulos, G., Galgani, L., Giorgilli, A.: On the number of isolating integrals in Hamiltonian systems. Phys. Rev. A 18, 1183–1189 (1978)ADSCrossRefGoogle Scholar
  35. 35.
    Devaney, R.L.: An introduction to chaotic dynamical systems. 2nd Ed. Addison-Wesley Publishing Company, New York (1989)MATHGoogle Scholar
  36. 36.
    Dieci, L., Van Vleck., E.S.: Computation of a few Lyapunov exponents for continuous and discrete dynamical systems. Appl. Num. Math. 17, 275–291 (1995)CrossRefMATHGoogle Scholar
  37. 37.
    Dieci, L., Van Vleck, E.S.: Lyapunov spectral intervals: theory and computation. SIAM J. Numer. Anal. 40, 516–542 (2002)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Dieci, L., Elia, C.: The singular value decomposition to approximate spectra of dynamical systems. Theoretical aspects. J. Diff. Eq. 230, 502–531 (2006)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Dieci, L., Lopez, L.: Smooth singular value decomposition on the symplectic group and Lyapunov exponents approximation. Calcolo 43, 1–15 (2006)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Dieci, L., Russell, R.D., Van Vleck, E.S.: On the computation of Lyapunov exponents for continuous dynamical systems. SIAM J. Numer. Anal. 34, 402–423 (1997)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Donnay, V.: Geodesic flow on the two-sphere, Part I: Positive measure entropy. Erg. Theory Dyn. Syst. 8, 531–553 (1988)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Donnay, V.: Geodesic flow on the two-sphere, Part II: Ergodicity. Lect. Notes Math. 1342, 112–153 (1988)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Donnay, V., Liverani, C.: Potentials on the two-torus for which the Hamiltonian flow is ergodic. Commun. Math. Phys. 135, 267–302 (1991)MathSciNetADSCrossRefMATHGoogle Scholar
  44. 44.
    Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)MathSciNetADSCrossRefGoogle Scholar
  45. 45.
    Eckmann, J.-P., Oliffson Kamphorst, S., Ruelle, D., Ciliberto, S.: Liapunov exponents from time series. Phys. Rev. A 34, 4971–4979 (1986)MathSciNetADSCrossRefGoogle Scholar
  46. 46.
    Farmer, J.D.: Chaotic attractors of an infinite-dimensional dynamical system. Physica D 4, 366–393 (1982)MathSciNetADSCrossRefMATHGoogle Scholar
  47. 47.
    Farmer, J.D., Ott, E., Yorke, J.A.: The dimension of chaotic attractors. Physica D 7, 153–180 (1983)MathSciNetADSCrossRefGoogle Scholar
  48. 48.
    Ford, J., Lunsford, G.H.: Stochastic behavior of resonant nearly linear oscillator systems in the limit of zero nonlinear coupling. Phys. Rev. A 1, 59–70 (1970)ADSCrossRefGoogle Scholar
  49. 49.
    Fouchard, M., Lega, E., Froeschlé, Ch., Froeschlé, C.: On the relationship between the fast Lyapunov indicator and periodic orbits for continuous flows. Cel. Mech. Dyn. Astron. 83, 205–222 (2002)ADSCrossRefMATHGoogle Scholar
  50. 50.
    Freistetter, F.: Fractal dimensions as chaos indicators. Cel. Mech. Dyn. Astron. 78, 211–225 (2000)MathSciNetADSCrossRefMATHGoogle Scholar
  51. 51.
    Froeschlé, C.: Numerical study of dynamical systems with three degrees of freedom II. Numerical displays of four-dimensional sections. Astron. Astrophs. 5, 177–183 (1970)ADSGoogle Scholar
  52. 52.
    Froeschlé, C.: Numerical study of a four-dimensional mapping. Astron. Astrophs. 16, 172–189 (1972)ADSGoogle Scholar
  53. 53.
    Froeschlé, C.: The Lyapunov characteristic exponents—Applications to celestial mechanics. Cel. Mech. Dyn. Astron. 34, 95–115 (1984)MATHGoogle Scholar
  54. 54.
    Froeschlé, C.: The Lyapunov characteristic exponents and applications. J. de Méc. Théor. et Appl. Numéro spécial 101–132 (1984)Google Scholar
  55. 55.
    Froeschlé, C.: The Lyapunov characteristic exponents and applications to the dimension of the invariant manifolds and chaotic attractors. In: Szebehely VG (ed.) Stability of the Solar System and Its Minor Natural and Artificial Bodies, 265–282, D. Reidel Publishing Company (1985)Google Scholar
  56. 56.
    Froeschlé, C., Lega, E.: On the structure of symplectic mappings. The fast Lyapunov indicator: a very sensitive tool. Cel. Mech. Dyn. Astron. 78, 167–195 (2000)ADSCrossRefMATHGoogle Scholar
  57. 57.
    Froeschlé, C., Froeschlé, Ch., Lohinger, E.: Generalized Lyapunov characteristic indicators and corresponding Kolmogorov like entropy of the standard mapping. Cel. Mech. Dyn. Astron. 56, 307–314 (1993)ADSCrossRefMATHGoogle Scholar
  58. 58.
    Froeschlé, C., Lega, E., Gonczi, R.: Fast Lyapunov indicators. Application to asteroidal motion. Cel. Mech. Dyn. Astron. 67, 41–62 (1997)ADSCrossRefMATHGoogle Scholar
  59. 59.
    Froeschlé, C., Gonczi, R., Lega, E.: The fast Lyapunov indicator: a simple tool to detect weak chaos. Application to the structure of the main asteroidal belt. Planet. Space Sci. 45, 881–886 (1997)ADSCrossRefGoogle Scholar
  60. 60.
    Frøyland, J.: Lyapunov exponents for multidimensional orbits. Phys. Let. A 97, 8–10 (1983)ADSCrossRefGoogle Scholar
  61. 61.
    Frøyland, J., Alfsen, K.H.: Lyapunov-exponent spectra for the Lorenz model. Phys. Rev. A 29, 2928–2931 (1984)ADSCrossRefGoogle Scholar
  62. 62.
    Geist, K., Parlitz, U., Lauterborn, W.: Comparison of different methods for computing Lyapunov exponents. Prog. Theor. Phys. 83, 875–893 (1990)MathSciNetADSCrossRefMATHGoogle Scholar
  63. 63.
    Goldhirsch, I., Sulem, P.-L., Orszag, S.A.: Stability and Lyapunov stability of dynamical systems: a differential approach and a numerical method. Physica D 27, 311–337 (1987)MathSciNetADSCrossRefMATHGoogle Scholar
  64. 64.
    Gottwald, G.A., Melbourne. I.: A new test for chaos in deterministic systems. Proc. Roy. Soc. London A 460, 603–611 (2004)MathSciNetADSCrossRefMATHGoogle Scholar
  65. 65.
    Gottwald, G.A., Melbourne, I.: Testing for chaos in deterministic systems with noise. Physica D 212, 100–110 (2005)MathSciNetADSCrossRefMATHGoogle Scholar
  66. 66.
    Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Physica D 9, 189–208 (1983)MathSciNetADSCrossRefMATHGoogle Scholar
  67. 67.
    Greene, J.M., Kim, J.-S.: The calculation of Lyapunov spectra. Physica D 24, 213–225 (1987)MathSciNetADSCrossRefMATHGoogle Scholar
  68. 68.
    Greub, W.: Multilinear Algebra. 2nd Ed. Springer, Berlin, Heidelberg, New York (1978)MATHGoogle Scholar
  69. 69.
    Guzzo, M., Lega, E., Froeschlé, C.: On the numerical detection of the effective stability of chaotic motions in quasi-integrable systems. Physica D 163, 1–25 (2002)MathSciNetADSCrossRefMATHGoogle Scholar
  70. 70.
    Haken, H.: At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point. Phys. Let. A 94, 71–72 (1983)MathSciNetADSCrossRefGoogle Scholar
  71. 71.
    Hegger, R., Kantz, H., Schreiber, T.: Practical implementation of nonlinear time series methods: The TISEAN package. Chaos 9, 413–435 (1999)ADSCrossRefMATHGoogle Scholar
  72. 72.
    Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73–79 (1964)ADSCrossRefGoogle Scholar
  73. 73.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)MATHGoogle Scholar
  74. 74.
    Howard, J.E. : Discrete virial theorem. Cel. Mech. Dyn. Astron. 92, 219–241 (2005)ADSCrossRefMATHGoogle Scholar
  75. 75.
    Hubbard, J.H., Hubbard, B.B.: Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach. Prentice Hall, New Jersey (1999)MATHGoogle Scholar
  76. 76.
    Johnson, B.A., Palmer, K.J., Sell, G.R.: Ergodic properties of linear dynamical systems. SIAM J. Math. Anal. 18, 1–33 (1987)MathSciNetCrossRefMATHGoogle Scholar
  77. 77.
    Kantz, H.: A robust method to estimate the maximal Lyapunov exponent of a time series. Phys. Let. A 185, 77–87 (1994)ADSCrossRefGoogle Scholar
  78. 78.
    Kantz, H., Schreiber, T.: Nonlinear time series analysis. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  79. 79.
    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, Lect. Notes Math. 730, 204–227 (1979)Google Scholar
  80. 80.
    Karanis, G.I., Vozikis, Ch.L.: Fast detection of chaotic behavior in galactic potentials. Astron. Nachr. 329, 403–412 (2008)ADSCrossRefGoogle Scholar
  81. 81.
    Kotoulas, T., Voyatzis, G.: Comparative study of the 2:3 and 3:4 resonant motion with Neptune: An application of symplectic mappings and low frequency analysis. Cel. Mech. Dyn. Astron. 88, 343–163 (2004)Google Scholar
  82. 82.
    Kovács, B.: About the efficiency of Fast Lyapunov Indicator surfaces and Small Alignment Indicator surfaces. PADEU, 19, 221–236 (2007)ADSGoogle Scholar
  83. 83.
    Laskar, J.: The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones. Icarus 88, 266–291 (1990)ADSCrossRefGoogle Scholar
  84. 84.
    Laskar, J.: Frequency analysis of multi-dimensional systems. Global dynamics and diffusion. Physica D 67, 257–281 (1993)MathSciNetADSCrossRefMATHGoogle Scholar
  85. 85.
    Laskar, J.: Introduction to frequency map analysis. In: Simó, C. (ed.): Hamiltonian systems with three or more degrees of freedom, 134–150, Plenum Press, New York (1999)Google Scholar
  86. 86.
    Laskar, J., Froeschlé, C., Celletti, A.: The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard map. Physica D 56, 253–269 (1992)MathSciNetADSCrossRefMATHGoogle Scholar
  87. 87.
    Ledrappier, F., Young, L.-S.: Dimension formula for random transformations. Commun. Math. Phys. 117, 529–548 (1988)MathSciNetADSCrossRefMATHGoogle Scholar
  88. 88.
    Lega, E., Froeschlé, C.: Comparison of convergence towards invariant distributions for rotation angles, twist angles and local Lyapunov characteristic numbers. Planet. Space Sci. 46, 1525–1534 (1998)ADSCrossRefGoogle Scholar
  89. 89.
    Lega, E., Froeschlé, C.: On the relationship between fast Lyapunov indicator and periodic orbits for symplectic mappings. Cel. Mech. Dyn. Astron. 81, 129–147 (2001)ADSCrossRefMATHGoogle Scholar
  90. 90.
    Li, C., Chen, G.: Estimating the Lyapunov exponents of discrete systems. Chaos 14, 343–346 (2004)MathSciNetADSCrossRefMATHGoogle Scholar
  91. 91.
    Li, C., Xia, X.: On the bound of the Lyapunov exponents of continuous systems. Chaos 14, 557–561 (2004)MathSciNetADSCrossRefMATHGoogle Scholar
  92. 92.
    Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. Second Edition. Springer, Berlin, Heidelberg, New York (1992)MATHGoogle Scholar
  93. 93.
    Lohinger, E., Froeschlé, C., Dvorak R.: Generalized Lyapunov exponents indicators in Hamiltonian dynamics: an application to a double star system. Cel. Mech. Dyn. Astron. 56, 315–322 (1993)ADSCrossRefMATHGoogle Scholar
  94. 94.
    Lu, J., Yang, G., Oh, H., Luo, A.C.J.: Computing Lyapunov exponents of continuous dynamical systems: method of Lyapunov vectors. Chaos Sol. Fract. 23, 1879–1892 (2005)MathSciNetMATHADSGoogle Scholar
  95. 95.
    Lukes-Gerakopoulos, G., Voglis, N., Efthymiopoulos, C.: The production of Tsallis entropy in the limit of weak chaos and a new indicator of chaoticity. Physica A 387, 1907–1925 (2008)ADSCrossRefMathSciNetGoogle Scholar
  96. 96.
    Lyapunov, A.M. (1992) The General Problem of the Stability of Motion. Taylor and Francis, London (English translation from the French: Liapounoff A (1907) Problème général de la stabilité du mouvement. Annal. Fac. Sci. Toulouse 9, 203–474. The French text was reprinted in Annals Math. Studies Vol. 17 Princeton Univ. Press (1947). The original was published in Russian by the Mathematical Society of Kharkov in 1892)Google Scholar
  97. 97.
    Markarian, R.: Non-uniformly hyperbolic billiards. Annal. Fac. Sci. Toulouse 3, 223–257 (1994)MathSciNetMATHGoogle Scholar
  98. 98.
    Mathiesen, J., Cvitanoviæ, P.: Lyapunov exponents. In: Cvitanoviæ, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G. (eds.): Chaos: Classical and Quantum. Niels Bohr Institute, Copenhagen, http://chaosbook.org/version12/ (2008)
  99. 99.
    Nagashima, T., Shimada, I.: On the C-system-like property of the Lorenz system. Prog. Theor. Phys. 58, 1318–1320 (1977)ADSCrossRefGoogle Scholar
  100. 100.
    Núñez, J.A., Cincotta, P.M., Wachlin, F.C.: Information entropy. An indicator of chaos. Cel. Mech. Dyn. Astron. 64, 43–53 (1996)ADSCrossRefMATHGoogle Scholar
  101. 101.
    Oliveira, S., Stewart, D.E.: Exponential splittings of products of matrices and accurately computing singular values of long products. Lin. Algebra Appl. 309, 175–190 (2000)MathSciNetCrossRefMATHGoogle Scholar
  102. 102.
    Oseledec, V.I.: A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231 (1968)MathSciNetGoogle Scholar
  103. 103.
    Ott, E.: Strange attractors and chaotic motions of dynamical systems. Rev. Mod. Phys. 53, 655–671 (1981)MathSciNetADSCrossRefMATHGoogle Scholar
  104. 104.
    Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S.: Geometry from time series. Phys. Rev. Let. 45, 712–716 (1980)ADSCrossRefGoogle Scholar
  105. 105.
    Paleari, S., Penati, S.: Numerical Methods and Results in the FPU Problem. Lect. Notes Phys. 728, 239–282 (2008)MathSciNetADSCrossRefGoogle Scholar
  106. 106.
    Pesin, Ya. B.: Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys 32, 55–114 (1977)MathSciNetCrossRefADSGoogle Scholar
  107. 107.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in Fortran. The Art of Scientific Computing. Cambridge University Press, Cambridge (2007)Google Scholar
  108. 108.
    Raghunathan, M.S.: A proof of Oseledec’s multiplicative ergodic theorem. Isr. J. Math. 32, 356–362 (1979)MathSciNetCrossRefMATHGoogle Scholar
  109. 109.
    Ramasubramanian, K., Sriram, M.S.: A comparative study of computation of Lyapunov spectra with different algorithms. Physica D 139, 72–86 (2000)MathSciNetADSCrossRefMATHGoogle Scholar
  110. 110.
    Rangarajan, G., Habib, S., Ryne, R.D.: Lyapunov exponents without rescaling and reorthogonalization. Phys. Rev. Let. 80, 3747–3750 (1998)ADSCrossRefGoogle Scholar
  111. 111.
    Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 65, 117–134 (1993)MathSciNetADSCrossRefMATHGoogle Scholar
  112. 112.
    Roux, J.-C., Simoyi, R.H., Swinney, H.L.: Observation of a strange attractor. Physica D 8, 257–266 (1983)MathSciNetADSCrossRefMATHGoogle Scholar
  113. 113.
    Ruelle, D.: Analycity properties of the characteristic exponents of random matrix products. Adv. Math. 32, 68–80 (1979)MathSciNetCrossRefMATHGoogle Scholar
  114. 114.
    Ruelle, D.: Ergodic theory of differentiable dynamical systems. IHES Publ. Math. 50, 275–306 (1979)Google Scholar
  115. 115.
    Sándor, Zs., Érdi, B., Efthymiopoulos, C.: The phase space structure around L4 in the restricted three-body problem. Cel. Mech. Dyn. Astron. 78, 113–123 (2000)ADSCrossRefMATHGoogle Scholar
  116. 116.
    Sándor Zs., Érdi, B., Széll, A., Funk, B.: The relative Lyapunov indicator: an efficient method of chaos detection. Cel. Mech. Dyn. Astron. 90, 127–138 (2004)ADSCrossRefMATHGoogle Scholar
  117. 117.
    Sandri, M.: Numerical calculation of Lyapunov exponents. Mathematica J. 6, 78–84 (1996)Google Scholar
  118. 118.
    Sano, M., Sawada, Y.: Measurement of the Lyapunov spectrum from a chaotic time series. Phys. Rev. Let. 55, 1082–1085 (1985)MathSciNetADSCrossRefGoogle Scholar
  119. 119.
    Shimada, I., Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical systems. Prog. Theor. Phys. 61, 1605–1615 (1979)MathSciNetADSCrossRefMATHGoogle Scholar
  120. 120.
    Sideris, I.V.: Characterization of chaos: a new, fast and effective measure. In: Gottesman, S.T., Buchler, J.-R. (eds.): Nonlinear Dynamics in Astronomy and Astrophysics, Annals of the New York Academy of Science, 1045:79, The New York Academy of Sciences (2005)Google Scholar
  121. 121.
    Sideris, I.V.: Measure of orbital stickiness and chaos strength. Phys. Rev. E 73, 066217 (2006)Google Scholar
  122. 122.
    Skokos Ch.: Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits. J. Phys. A 34, 10029–10043 (2001)MathSciNetADSCrossRefMATHGoogle Scholar
  123. 123.
    Skokos, Ch., Contopoulos, G., Polymilis, C.: Structures in the phase space of a four dimensional symplectic map. Cel. Mech. Dyn. Astron. 65, 223–251 (1997)MathSciNetADSCrossRefMATHGoogle Scholar
  124. 124.
    Skokos, Ch., Antonopoulos, Ch., Bountis, T.C., Vrahatis, M.N.: How does the smaller alignment index (SALI) distinguish order from chaos? Prog. Theor. Phys. Supp. 150, 439–443 (2003)ADSCrossRefGoogle Scholar
  125. 125.
    Skokos, Ch., Antonopoulos, Ch., Bountis, T.C., Vrahatis, M.N.: Detecting order and chaos in Hamiltonian systems by the SALI method. J. Phys. A 37, 6269–6284 (2004)MathSciNetADSCrossRefGoogle Scholar
  126. 126.
    Skokos, Ch., Bountis, T.C., Antonopoulos, Ch.: Geometrical properties of local dynamics in Hamiltonian systems: The generalized alignment index (GALI) method. Physica D 231, 30–54 (2007)MathSciNetADSCrossRefMATHGoogle Scholar
  127. 127.
    Skokos, Ch., Bountis, T.C., Antonopoulos, Ch.: Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi-Pasta-Ulam lattices by the generalized alignment index method. Eur. Phys. J. Sp. Top. 165, 5–14 (2008)CrossRefGoogle Scholar
  128. 128.
    Spivak, M.: Calculus on Manifolds. Addison-Wesley, New York (1965)MATHGoogle Scholar
  129. 129.
    Spivak, M.: Comprehensive Introduction to Differential Geometry, vol. 1. Publish or Perish Inc., Houston (1999)Google Scholar
  130. 130.
    Stewart, D.E.: A new algorithm for the SVD of a long product of matrices and the stability of products. Electr. Trans. Numer. Anal. 5, 29–47 (1997)MATHGoogle Scholar
  131. 131.
    Stoddard, S.D., Ford, J.: Numerical experiments on the stochastic behavior of a Lennard–Jones gas system. Phys. Rev. A 8, 1504–1512 (1973)ADSCrossRefGoogle Scholar
  132. 132.
    Süli, Á.: Speed and efficiency of chaos detection methods. In: Süli, Á., Freistetter, F., Pál, A. (eds.): Proceedings of the 4th Austrian Hungarian workshop on Celestial Mechanics, 18, 179–189, Publications of the Astronomy Department of the Eötvös University (2006)Google Scholar
  133. 133.
    Süli, Á.: Motion indicators in the 2D standard map. PADEU 17, 47–62 (2006)ADSGoogle Scholar
  134. 134.
    Takens, F.: Detecting strange attractors in turbulence. Lect. Notes Math. 898, 366–381 (1981)MathSciNetCrossRefGoogle Scholar
  135. 135.
    Voglis, N., Contopoulos, G.: Invariant spectra of orbits in dynamical systems. J. Phys. A 27, 4899–4909 (1994)MathSciNetADSCrossRefMATHGoogle Scholar
  136. 136.
    Voglis, N., Contopoulos, G., Efthymiopoulos, C.: Method for distinguishing between ordered and chaotic orbits in four-dimensional maps. Phys. Rev. E 57, 372–377 (1998)ADSCrossRefGoogle Scholar
  137. 137.
    Voyatzis, G., Ichtiaroglou, S.: On the spectral analysis of trajectories in near-integrable Hamiltonian systems. J. Phys. A 25, 5931–5943 (1992)MathSciNetADSCrossRefMATHGoogle Scholar
  138. 138.
    Vozikis, Ch.L., Varvoglis, H., Tsiganis, K.: The power spectrum of geodesic divergences as an early detector of chaotic motion. Astron. Astrophs. 359, 386–396 (2000)ADSGoogle Scholar
  139. 139.
    Vrahatis, M.N., Bountis, T.C., Kollmann, M.: Periodic orbits and invariant surfaces of 4D nonlinear mappings. Int. J. Bif. Chaos 6, 1425–1437 (1996)MathSciNetCrossRefMATHGoogle Scholar
  140. 140.
    Vrahatis, M.N., Isliker, H., Bountis, T.C.: Structure and breakdown of invariant tori in a 4-D mapping model of accelerator dynamics. Int. J. Bif. Chaos 7, 2707–2722 (1997)CrossRefMATHGoogle Scholar
  141. 141.
    Walters, P.: A dynamical proof of the multiplicative ergodic theorem. Thans. Amer. Math. Soc. 335, 245–257 (1993)MathSciNetCrossRefMATHGoogle Scholar
  142. 142.
    Wojtkowski, M.: Invariant families of cones and Lyapunov exponents. Erg. Theory Dyn. Syst. 5, 145–161 (1985)MathSciNetCrossRefMATHGoogle Scholar
  143. 143.
    Wojtkowski, M.: Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys. 105, 391–414 (1986)MathSciNetADSCrossRefMATHGoogle Scholar
  144. 144.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)MathSciNetADSCrossRefMATHGoogle Scholar
  145. 145.
    Wu, X., Huang, T.-Y., Zhang, H.: Lyapunov indices with two nearby trajectories in a curved spacetime. Phys. Rev. E 74:083001 (2006)MathSciNetGoogle Scholar
  146. 146.
    Young, L.-S.: Dimension, entropy and Lyapunov exponents. Erg. Theory Dyn. Syst. 2, 109–124 (1982)CrossRefMATHGoogle Scholar
  147. 147.
    Zou, Y., Pazó, D., Romano, M.C., Thiel, M., Kurths, J.: Distinguishing quasiperiodic dynamics from chaos in short-time series. Phys. Rev. E 76:016210 (2007)MathSciNetADSCrossRefGoogle Scholar
  148. 148.
    Zou, Y., Thiel, M., Romano, M.C., Kurths, J.: Characterization of stickiness by means of recurrence. Chaos 17:043101 (2007)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Astronomie et Systèmes DynamiquesIMCCE, Observatoire de ParisParisFrance
  2. 2.Max Planck Institute for the Physics of Complex SystemsDresdenGermany

Personalised recommendations