Journal of Statistical Physics

, Volume 146, Issue 1, pp 56–66 | Cite as

Accelerating Cycle Expansions by Dynamical Conjugacy

  • Ang Gao
  • Jianbo Xie
  • Yueheng Lan


Periodic orbit theory provides two important functions—the dynamical zeta function and the spectral determinant for the calculation of dynamical averages in a nonlinear system. Their cycle expansions converge rapidly when the system is uniformly hyperbolic but greatly slow down in the presence of non-hyperbolicity. We find that the slow convergence can be attributed to singularities in the natural measure. A properly designed coordinate transformation may remove these singularities and results in a dynamically conjugate system where fast convergence is restored. The technique is successfully demonstrated on several examples of one-dimensional maps and some remaining challenges are discussed.


Cycle expansions Periodic orbit theory Nonlinear dynamics Nonequilibrium statistical physics Dynamical zeta function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Artuso, R., Aurell, E., Cvitanović, P.: Recycling of strange sets: I. Cycle expansions. Nonlinearity 3, 325 (1990) CrossRefMATHADSMathSciNetGoogle Scholar
  2. 2.
    Artuso, R., Aurell, E., Cvitanović, P.: Recycling of strange sets: II. Applications. Nonlinearity 3, 361 (1990) CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Artuso, R., Cvitanović, P., Tanner, G.: Cycle expansions for intermittent maps. Proc. Theor. Phys. Suppl. 150, 1–21 (2003) CrossRefADSGoogle Scholar
  4. 4.
    Aurell, E.: Convergence of dynamical zeta functions. J. Stat. Phys. 58, 967 (1990) CrossRefMATHADSMathSciNetGoogle Scholar
  5. 5.
    Belkić, D., Main, J., Dando, P.A., Taylor, H.S.: Semiclassical quantization by Padé approximant to periodic orbit sums. Europhys. Lett. 48, 250 (1999) CrossRefADSGoogle Scholar
  6. 6.
    Cvitanović, P.: Invariant measurement of strange sets in terms of cycles. Phys. Rev. Lett. 61, 2729 (1988) CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Cvitanović, P.: Universality in Chaos, 2nd edn. Hilger, Bristol (1989) MATHGoogle Scholar
  8. 8.
    Cvitanović, P.: Periodic orbits as the skeleton of classical and quantum chaos. Physica D 51, 138 (1991) CrossRefMATHADSMathSciNetGoogle Scholar
  9. 9.
    Cvitanović, P., Hansen, K., Rolf, J., Vattay, G.: Beyond the periodic orbit theory. Nonlinearity 11, 1209 (1998) CrossRefMATHADSMathSciNetGoogle Scholar
  10. 10.
    Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G.: Chaos: Classical and Quantum. Niels Bohr Institute, Copenhagen (2005). Google Scholar
  11. 11.
    Dahlqvist, P.: On the effect of pruning on the singularity structure of zeta functions. J. Math. Phys. 38, 4273 (1997) CrossRefMATHADSMathSciNetGoogle Scholar
  12. 12.
    Dettmann, C.P., Cvitanović, P.: Cycle expansions for intermittent diffusion. Phys. Rev. E 56, 6687 (1997) CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Eckhardt, B., Russberg, G.: Resummation of classical and semiclassical periodic-orbit formulas. Phys. Rev. E 47, 1578 (1993) CrossRefADSGoogle Scholar
  14. 14.
    Frisch, U.: Turbulence. Cambridge University Press, Cambridge (1996) Google Scholar
  15. 15.
    Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Springer, New York (1990) MATHGoogle Scholar
  16. 16.
    Hao, B.-L.: Chaos, vol. II. World Scientific, Singapore (1990) Google Scholar
  17. 17.
    Hatjispyros, S., Vivaldi, F.: A family of rational zeta functions for the quadratic map. Nonlinearity 8, 321 (1995) CrossRefMATHADSMathSciNetGoogle Scholar
  18. 18.
    Nielsen, S.F., Dahlqvist, P., Cvitanović, P.: Periodic orbit sum rules for billiards: accelerating cycle expansions. J. Phys. A, Math. Gen. 32, 6757 (1999) CrossRefMATHADSGoogle Scholar
  19. 19.
    Quyen, M.L.V., Martinerie, J., Adam, C., Varela, F.J.: Unstable periodic orbits in human epileptic activity. Phys. Rev. E 56, 3401 (1997) CrossRefADSGoogle Scholar
  20. 20.
    Rugh, H.H.: The correlation spectrum for hyperbolic analytic maps. Nonlinearity 5, 1237 (1992) CrossRefMATHADSMathSciNetGoogle Scholar
  21. 21.
    Sinai, Y.G.: Introduction to Ergodic Theory. Princeton University Press, Princeton (1976) MATHGoogle Scholar
  22. 22.
    So, P., Francis, J.T., Netoff, T.I., Gluckman, B.J., Sciff, S.J.: Periodic orbits: a new language for neuronal dynamics. Biophys. J. 74, 2776 (1998) CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.The Department of PhysicsTsinghua UniversityBeijingChina
  2. 2.The Department of PhysicsUC BerkeleyBerkeleyUSA

Personalised recommendations