Stochastic Approach to Lyapunov Exponents in Coupled Chaotic Systems

  • Rüdiger Zillmer
  • Volker Ahlers
  • Arkady Pikovsky
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 557)


We use linear stochastic Langevin equations to describe the Lyapunov exponents in coupled chaotic systems. The largest Lyapunov exponent is calculated analytically, using the stationary solution of the Fokker-Planck equation. For small couplings we reproduce the singularity which was first described by Daido as the effect of coupling sensitivity of chaos. The singularity is shown to depend on the coupling and the systems’ mismatch. The analytical results are confirmed by numerical simulations.


Lyapunov Exponent Chaotic System Large Lyapunov Exponent Logarithmic Singularity Small Coupling 
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  1. 1.
    A. S. Pikovsky. On the interaction of strange attractors. Z. Physik B, 55(2):149–154, 1984.CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    A. S. Pikovsky and P. Grassberger. Symmetry breaking bifurcation for coupled chaotic attractors. J. Phys. A: Math., Gen., 24(19):4587–4597, 1991.CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    R. Zillmer, V. Ahlers, and A. Pikovsky. Scaling of Lyapunov exponents of coupled chaotic systems. Phys. Rev. E, 61(1):332–341, 2000.CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    H. Daido. Coupling sensitivity of chaos. Prog. Theor. Phys., 72(4):853–856, 1984.CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    H. Daido. Coupling sensitivity of chaos. Prog. Theor. Phys. Suppl., 79:75–95, 1984.CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    H. Daido. Coupling sensitivity of chaos and the Lyapunov dimension, the case of coupled two-dimensional maps. Phys. Lett. A, 110:5, 1985.CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    R. Livi, A. Politi, and S. Ruffo. Scaling law for the maximal Lyapunov exponent. J. Phys. A: Math., Gen., 25:4813, 1992.zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    A. Torcini, R. Livi, A. Politi, and S. Ruffo. Comment on "Universal scaling law for the largest Lyapunov exponent in coupled map lattices". Phys. Rev. Lett., 78(7):1391, 1997.CrossRefADSGoogle Scholar
  9. 9.
    F. Cecconi and A. Politi. Analytic estimate of the maximum Lyapunov exponent in products of tridiagonal random matrices. J. Phys. A: Math., Gen., 32(44): 7603–7622, 1999.zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada. Intermittency caused by chaotic modulation II. Prog. Theor. Phys., 76(6):1198–1209, 1986.CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    H. Z. Risken. The Fokker-Planck Equation. Springer, Berlin, 1989.zbMATHGoogle Scholar
  12. 12.
    A. Crisanti, G. Paladin, and A. Vulpiani. Products of Random Matrices in Statistical Physics. Springer, Berlin, 1993.zbMATHGoogle Scholar
  13. 13.
    L. A. Bunimovich and Ya. G. Sinai. Spacetime chaos in coupled map lattices. Nonlinearity, 1(4):491–516, 1988.zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Department of Commerce USA, Washington, D.C., 1964.zbMATHGoogle Scholar
  15. 15.
    C. W. Gardiner. Handbook of Stochastic Methods. Springer, Berlin, 1996.Google Scholar
  16. 16.
    K. Geist, U. Parlitz, and W. Lauterborn. Comparison of different methods for computing Lyapunov exponents. Prog. Theor. Phys., 83:875–893, 1991.CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    A. Bjorck. Linear Algebra Appl., 197/198:297, 1994.CrossRefMathSciNetGoogle Scholar
  18. 18.
    J. P. Eckmann and I. Procaccia. Fluctuations of dynamical scaling indices in nonlinear systems. Phys. Rev. A, 34(l):659–663, 1986.CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Rüdiger Zillmer
    • 1
  • Volker Ahlers
    • 1
  • Arkady Pikovsky
    • 1
  1. 1.Department of PhysicsUniversity of PotsdamPotsdamGermany

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