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Nonlinear Dynamics

, Volume 61, Issue 1–2, pp 241–249 | Cite as

Dynamical randomicity and predictive analysis in cubic chaotic system

  • Yagang Zhang
  • Yan Xu
  • Zengping Wang
Original Paper

Abstract

The dynamical randomicity and grey prediction in cubic chaotic system will be studied in this paper. These stochastic symbolic sequences bear three features. The distribution of frequency, inter-occurrence times, the first passage time, the rth passage time and the ordinal passage time are discussed separately. By using transfer probability of Markov chain (MC), one obtains analytic expressions of generating functions in three-probabilities stochastic wander model, which can be applied to all three symbolic systems. The visitation density function of cubic map will also be resolved. Especially, after introducing grey system theory, one is mainly using GM(1,1) model to forecast data sequences, and the usual forecast precision is approximately 90%. In the symbolic prediction of cubic chaotic dynamical system, the precision of grey prediction certainly will decrease as the length of symbolic sequence is increasing. But in this place we have found a generating rule that may realize chaotic synchronization at least in short and medium terms, and we can analyze and forecast in this way.

Symbolic dynamics Cubic chaotic system First passage time Grey prediction 

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References

  1. 1.
    Ulam, S.M., Neumann, J.V.: On combination of stochastic and deterministic processes. Bull. Am. Math. Soc. 53, 1120 (1947) Google Scholar
  2. 2.
    Hao, B.L., Zheng, W.M.: Symbolic Dynamics and Chaos. Directions in Chaos. World Scientific, Singapore (1998) Google Scholar
  3. 3.
    Collet, P., Eckmann, J.P.: Iterated Maps on the Interval as Dynamical Systems. Birkhäuser, Boston (1980) MATHGoogle Scholar
  4. 4.
    Peng, S.L., Luo, L.S.: The ordering of critical periodic points in coordinate space by symbolic dynamics. Phys. Lett. A 153, 345–352 (1991) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Zhang, Y.G., Wang, C.J., Zhou, Z.: Inherent randomicity in 4-symbolic dynamics. Chaos Solitons Fractals 28, 236–243 (2006) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Zhang, Y.G., Wang, C.J.: Multiformity of inherent randomicity and visitation density in n symbolic dynamics. Chaos Solitons Fractals 33, 685–694 (2007) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Deng, J.: Introduction to grey system theory. J. Grey Syst. 1, 1–24 (1989) MATHGoogle Scholar
  8. 8.
    Deng, J.: Control problems of grey system. Syst. Control Lett. 5, 288–294 (1982) Google Scholar
  9. 9.
    Lin, C.T., Yang, S.Y.: Forecast of the output value of Taiwan’s optoelectronics industry using the grey forecasting model. Technol. Forecast. Soc. Change 70, 177–186 (2003) CrossRefGoogle Scholar
  10. 10.
    Akay, D., Atak, M.: Grey prediction with rolling mechanism for electricity demand forecasting of Turkey. Energy 32, 1670–1675 (2007) CrossRefGoogle Scholar
  11. 11.
    Mao, M., Chirwa, E.C.: Application of grey model GM(1,1) to vehicle fatality risk estimation. Technol. Forecast. Soc. Change 73, 588–605 (2006) CrossRefGoogle Scholar
  12. 12.
    Li, P.L.: Grey forecasting of earthquake (M greater than or equal to 6.0) frequency in China mainland. J. Grey Syst. 2, 133–151 (1990) Google Scholar
  13. 13.
    Zhang, Y.G., Xu, Y., Wang, Z.P.: GM(1,1) grey prediction of Lorenz chaotic system. Chaos Solitons Fractals 42, 1003–1009 (2009) CrossRefGoogle Scholar
  14. 14.
    Tseng, F.M., Yu, H.C., Tzeng, G.H.: Applied hybrid grey model to forecast seasonal time series. Technol. Forecast. Soc. Change 70, 563–574 (2003) CrossRefGoogle Scholar
  15. 15.
    Liu, S., Lin, Y.: Grey Information: Theory and Practical Applications. Springer, Berlin (2006) Google Scholar
  16. 16.
    Chiang, J.Y., Chen, C.K.: Application of grey prediction to inverse nonlinear heat conduction problem. Int. J. Heat Mass Transfer 51, 576–585 (2008) MATHCrossRefGoogle Scholar
  17. 17.
    Peng, S.L., Zhang, X.S., Cao, K.F.: Dual star products and metric universality in symbolic dynamics of three letters. Phys. Lett. A 246, 87–96 (1998) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zeng, W.Z., Ding, M.Z., Li, J.N.: Symbolic description of periodic windows in the antisymmetric cubic map. Chin. Phys. Lett. 2, 293–296 (1985) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Zeng, W.Z., Ding, M.Z., Li, J.N.: Symbolic dynamics for one-dimensional mappings with multiple critical points. Commun. Theor. Phys. 9, 141–152 (1988) MathSciNetGoogle Scholar
  20. 20.
    Fang, H.P.: Dynamics for a two-dimensional antisymmetric map. J. Phys. A, Math. Gen. 27, 5187–5200 (1994) MATHCrossRefGoogle Scholar
  21. 21.
    Benediktovitch, I., Romanovski, V.: Bautin ideal of a cubic map. Appl. Math. Lett. 14, 159–165 (2001) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Coelho, Z., Collet, P.: Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle. J. Probab. Theory Relat. Fields 99, 237–250 (1994) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Coelho, Z.: On discrete stochastic processes generated by deterministic sequences and multiplication machines. Indag. Math. NS 11, 359–378 (2000) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Liang, X., Jiang, J.F.: On the topological entropy, non-wandering set and chaos of monotone and competitive dynamical systems. Chaos Solitons Fractals 14, 689–696 (2002) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Peng, S.L., Cao, K.F., Chen, Z.X.: Devil’s staircase of topological entropy and global metric regularity. Phys. Lett. A 193, 437–443 (1994) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Ding, J., Li, T.Y.: Markov finite approximation of Frobenius–Perron operator. Nonlinear Anal. Theory Methods Appl. 17, 759–772 (1991) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Li, T.Y.: Finite approximation for the Frobenius–Perron operator, a solution to Ulam’s conjecture. J. Approx. Theory 17, 177–186 (1976) MATHCrossRefGoogle Scholar
  28. 28.
    Lasota, A., Mackey, M.C.: Probabilistic Properties of Deterministic Systems. Cambridge University Press, Cambridge (1985) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsNorth China Electric Power UniversityBaodingPeople’s Republic of China
  2. 2.Key Laboratory of Power System Protection and Dynamic Security Monitoring and Control under Ministry of EducationNorth China Electric Power UniversityBaodingPeople’s Republic of China
  3. 3.Center for Nonlinear Complex Systems, School of Physical Science and TechnologyYunnan UniversityKunmingPeople’s Republic of China

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