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Stochastic Control Part 3

Tham khảo tài liệu 'stochastic control part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 72 Stochastic Control Fig. 10. Mean density p vs. p for different rules evolving under the third self-synchronization method. The density of the system decreases linearly with p. that the behavior reported in the first self-synchronization method is newly obtained in this case. Rule 18 undergoes a phase transition for a critical value of p. For p greater than the critical value the method is able to find the stable structure of the system Sanchez Lopez-Ruiz 2006 . For the rest of the rules the freezing phase is not found. The dynamics generates patterns where the different marginally stable structures randomly compete. Hence the DA density decays linearly with p see Fig. 8 . 4.3 Third Self-Synchronization Method At last we introduce another type of stochastic element in the application of the rule . Given an integer number L the surrounding of site i at each time step is redefined. A site ii is randomly chosen among the L neighbors of site i to the left i L . i 1 . Analogously a site ir is randomly chosen among the L neighbors of site i to the right i 1 . i L . The rule is now applied on the site i using the triplet ii i ir instead of the usual nearest neighbors of the site. This new version of the rule is called L being 0L 1 . Later the operator rp acts in identical way as in the first method. Therefore the dynamical evolution law is a t 1 Tp a1 t a2 t Tp a t L a t . 13 The DA density as a function of p is plotted in Fig. 9 for the rule 18 and in Fig. 10 for other rules. It can be observed again that the rule 18 is a singular case that even for different L maintains the memory and continues to self-synchronize. It means that the influence of the rule is even more important than the randomness in the election of the surrounding sites. The system self-synchronizes and decays to the corresponding stable structure. Contrary for the rest of the rules the DA density decreases linearly with p even for L 1 as shown in Fig. 10. Complexity and stochastic synchronization in