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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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)