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.
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References
Ulam, S.M., Neumann, J.V.: On combination of stochastic and deterministic processes. Bull. Am. Math. Soc. 53, 1120 (1947)
Hao, B.L., Zheng, W.M.: Symbolic Dynamics and Chaos. Directions in Chaos. World Scientific, Singapore (1998)
Collet, P., Eckmann, J.P.: Iterated Maps on the Interval as Dynamical Systems. Birkhäuser, Boston (1980)
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)
Zhang, Y.G., Wang, C.J., Zhou, Z.: Inherent randomicity in 4-symbolic dynamics. Chaos Solitons Fractals 28, 236–243 (2006)
Zhang, Y.G., Wang, C.J.: Multiformity of inherent randomicity and visitation density in n symbolic dynamics. Chaos Solitons Fractals 33, 685–694 (2007)
Deng, J.: Introduction to grey system theory. J. Grey Syst. 1, 1–24 (1989)
Deng, J.: Control problems of grey system. Syst. Control Lett. 5, 288–294 (1982)
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)
Akay, D., Atak, M.: Grey prediction with rolling mechanism for electricity demand forecasting of Turkey. Energy 32, 1670–1675 (2007)
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)
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)
Zhang, Y.G., Xu, Y., Wang, Z.P.: GM(1,1) grey prediction of Lorenz chaotic system. Chaos Solitons Fractals 42, 1003–1009 (2009)
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)
Liu, S., Lin, Y.: Grey Information: Theory and Practical Applications. Springer, Berlin (2006)
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)
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)
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)
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)
Fang, H.P.: Dynamics for a two-dimensional antisymmetric map. J. Phys. A, Math. Gen. 27, 5187–5200 (1994)
Benediktovitch, I., Romanovski, V.: Bautin ideal of a cubic map. Appl. Math. Lett. 14, 159–165 (2001)
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)
Coelho, Z.: On discrete stochastic processes generated by deterministic sequences and multiplication machines. Indag. Math. NS 11, 359–378 (2000)
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)
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)
Ding, J., Li, T.Y.: Markov finite approximation of Frobenius–Perron operator. Nonlinear Anal. Theory Methods Appl. 17, 759–772 (1991)
Li, T.Y.: Finite approximation for the Frobenius–Perron operator, a solution to Ulam’s conjecture. J. Approx. Theory 17, 177–186 (1976)
Lasota, A., Mackey, M.C.: Probabilistic Properties of Deterministic Systems. Cambridge University Press, Cambridge (1985)
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Zhang, Y., Xu, Y. & Wang, Z. Dynamical randomicity and predictive analysis in cubic chaotic system. Nonlinear Dyn 61, 241–249 (2010). https://doi.org/10.1007/s11071-009-9645-2
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DOI: https://doi.org/10.1007/s11071-009-9645-2