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A novel construction of chaotic dynamics base gliders in diffusion rule B2/S7

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Abstract

The dynamical behaviors of gliders (mobile localizations) in diffusion rule B2/S7 are quantitatively analyzed from the theory of symbolic dynamics in two-dimensional symbolic sequence space. Their intrinsic complexity is demonstrated by exploiting the relationship between one-dimensional and two-dimensional subshifts. Based on this rigorous approach and technique, the underlying chaos of the extant gliders and their combinations is characterized in a subtle way. It is demonstrated that they can be identified to distinct subsystems with very rich and complicated dynamics; that is, diffusion rule is topologically mixing and possesses positive topological entropy on each subsystem. This analytical assertion provides the fact that diffusion rule is covered with complex subsystems “almost everywhere”. Finally, it is worth mentioning that the procedure proposed in this paper is also applicable to all other gliders arising from the two-dimensional cellular automata therein. It is an extended discovery in both cellular automata and chaos theory.

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References

  1. Peng, G., Nie, F., Cao, B., Liu, C.: A driver’s memory lattice model of traffic flow and its numerical simulation. Nonlinear Dyn. 67(3), 1811–1815 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Goldstein, J.: The singular nature of emergent levels: suggestions for a theory of emergence. Nonlinear Dyn. 6(4), 293–309 (2002)

    Google Scholar 

  3. Taniguchi, Y., Suzuki, H.: A traffic cellular automaton with estimation of time to collision. J. Cell. Autom. 5–6, 407–416 (2013)

    MathSciNet  Google Scholar 

  4. Adamatzky, A., Wuensche, A., Benjamin, D.: Glider-based computing in reaction-diffusion hexagonal cellular automata. Chaos Sol. Fract. 27, 287–295 (2006)

    Article  MATH  Google Scholar 

  5. Neumann, J.V., Burks, A.W.: The Theory of Self-Reproducing Automata. University of Illinois Press, Urbana (1996)

    Google Scholar 

  6. Gardner, M.: The fantastic combinations of John Conway’s new solitaire game ‘life’. Sci. Am. 223, 120–123 (1970)

    Article  Google Scholar 

  7. Adamatzky, A.: Game of Life Cellular Automata. Springer, London (2010)

    Book  MATH  Google Scholar 

  8. Hedlund, G.A.: Endomorphisms and automorphism of the shift dynamical system. Theory Comput. Syst. 3, 320–375 (1969)

    MATH  MathSciNet  Google Scholar 

  9. Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601–644 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  10. Wolfram, S.: Theory and Applications of Cellular Automata. World Scientifc, Singapore (1986)

    MATH  Google Scholar 

  11. Li, W., Packard, N.: The structure of elementary cellular automata rule space. Complex Syst. 4, 281–297 (1990)

    MathSciNet  Google Scholar 

  12. Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign (2002)

  13. Chua, L.O., Sbitnev, V.I., Yoon, S.: A nonlinear dynamics perspective of Wolfram’s new kind of science. Part IV: from bernoulli-shift to 1/f spectrum. Int. J. Bifurc Chaos 15(4), 1045–1223 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chua, L.O., Pazienza, G.E.: A nonlinear dynamics perspective of Wolfram’s new kind of science. Part XIII: Bernoulli \(\sigma _\tau \)-shift rules. Int. J. Bifurc Chaos 20(7), 1859–2003 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chua, L.O., Pazienza, G.E.: A nonlinear dynamics perspective of Wolfram’s new kind of science. Part XIV: More Bernoulli \(\sigma _\tau \)-shift rules. Int. J. Bifurc Chaos 20(8), 2253–2425 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Chua, L.O., Sbitnev, V.I., Yoon, S.: A nonlinear dynamics perspective of Wolfram’s new kind of science. Part VI: from time-reversible attractors to the arrows of time. Int. J. Bifurc Chaos 16(5), 1097–1373 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Wuensche, A., Lesser, M.J.: The Global Dynamics of Cellular Automata: An Atlas of Basin of Attraction Fields of One-Dimensional Cellular Automata. Addison-Wesley, Reading (1992)

    MATH  Google Scholar 

  18. Cattaneo, G., Formenti, E., Margara, L., Mauri, G.: On the dynamical behavior of chaotic cellular automata. Theory Comput. Sci. 217, 31–51 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jin, W.F., Chen, F.Y., Chen, G.R., Chen, L., Chen, F.F.: Extending the symbolic dynamics of Chua’s Bernoulli-shift rule 56. J. Cell. Autom. 5(1–2), 121–138 (2010)

    MATH  MathSciNet  Google Scholar 

  20. Chen, F.Y., Jin, W.F., Chen, G.R., Chen, F.F., Chen, L.: Chaos of elementary cellular automata rule 42 of Wolfram’s class II. Chaos 19(1), 013140 (2009)

  21. Chen, G.R., Chen, F.Y., Guan, J.B., Jin. W.F.: Symbolic dynamics of some bernoulli-shift cellular automata rules. 2010 International Symposium on Nonlinear Theory and its Applications, pp. 595–598 (2010)

  22. Martinez, G.J., Adamatzky, A., McIntosh, H.V.: Localization dynamics in a binary two-dimensional cellular automaton: the diffusion rule. In: Adamatzky, A. (ed.) Game of Life Cellular Automata, pp. 291–315. Springer, London (2010)

    Chapter  Google Scholar 

  23. Cook, M.: Universality in elementary cellular automata. Complex Syst. 15(1), 1–40 (2004)

    MATH  Google Scholar 

  24. Adamatzky, A., Wuensche, A.: Computing in spiral rule reaction-diffusion hexagonal cellular automaton. Complex Syst. 16(4), 277–297 (2006)

    MathSciNet  Google Scholar 

  25. Jin, W.F., Chen, F.Y., Chen, G.R.: Glider implies Li–Yorke chaos for one-dimensional cellular automata. J. Cell. Autom. (in press)

  26. Cattaneo, G., Finelli, M., Margara, L.: Investigating topological chaos by elementary cellular automata dynamics. Theo. Comput. Sci. 1–2, 219–241 (2000)

    Article  MathSciNet  Google Scholar 

  27. Lind, D.: Multi-dimensional symbolic dynamics. In: Proceedings of Symposia in Applied Mathematics (2004)

  28. Kitchens, B.: Symbolic Dynamics: One-Sided, Two-Sided and Countable State Markov Shifts. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  29. Blanchard, F., Glasner, E., Kolyada, S.: On Li–Yorke pairs. J. Reine Angew. Math. 547, 51–68 (2002)

    MATH  MathSciNet  Google Scholar 

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Acknowledgments

This research was jointly supported by NSFC (Grants no. 11171084 and 60872093), and Foundation of Zhejiang Chinese Medical University (Grant no. 17211076).

Author information

Authors and Affiliations

  1. College of Pharmaceutical Sciences, Zhejiang Chinese Medical University, Hangzhou, Zhejiang, China

    Weifeng Jin

  2. School of Science, Shanghai University, Shanghai, China

    Weifeng Jin

  3. School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, China

    Fangyue Chen

Authors
  1. Weifeng Jin
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  2. Fangyue Chen
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Corresponding author

Correspondence to Weifeng Jin.

Appendix

Appendix

  1. 1.

    The determinative block system \(\fancyscript{A}_1\) is \(\fancyscript{A}_1=\fancyscript{A}=\{(A_0,A_0,A_0), \ (A_0,A_0,A_1),\ (A_0,A_1,A_2)\), \((A_1,A_2,A_0), (A_2,A_0,A_0)\}\), and its transition matrix \(A_1\) is \(A_1=A=\left[ \begin{array}{ccccc} 1&{}\ 1&{}\ 0&{}\ 0&{}\ 0 \\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0 \\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0 \\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1\\ 1&{}\ 1&{}\ 0&{}\ 0&{}\ 0 \end{array} \right] \).

  2. 2.

    The determinative block system \(\fancyscript{A}_2\) is \(\fancyscript{A}_2=\{(A_3,A_4,A_2), (A_4,A_2,A_0), (A_2,A_0,A_0)\), \( (A_0,A_0,A_3)\), \((A_0,A_3,A_4), (A_0,A_0,A_0)\}\), and its transition matrix \(A_2\) is \(A_2\!=\!\left[ \! \begin{array}{cccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ \end{array} \!\right] \!\).

  3. 3.

    The determinative block system \(\fancyscript{A}_3\) is \(\fancyscript{A}_1=\{(A_4,A_3,A_2), \ (A_3,A_2,A_0), \ (A_2,A_0,A_0)\), \((A_0,A_0, A_4)\), \((A_0,A_4,A_3),(A_0,A_0,A_0)\}\), and its transition matrix \(A_3\) is \(A_3=\left[ \begin{array}{cccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ \end{array} \right] \).

  4. 4.

    The determinative block system \(\fancyscript{A}_4\) is \(\fancyscript{A}_4=\{(A_1,A_0,A_2), \ (A_0,A_2,A_0), \ (A_2,A_0,A_0)\), \((A_0,A_0,A_1)\), \((A_0,A_1,A_0),(A_0,A_0,A_0)\}\), and its transition matrix \(A_4\) is \(A_4=\left[ \begin{array}{cccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ \end{array} \right] \).

  5. 5.

    The determinative block system \(\fancyscript{A}_5\) is \(\fancyscript{A}_5=\fancyscript{B}=\{(A_0,A_0,A_0), (A_0,A_0,A_1), (A_0,A_1,A_2), (A_1,A_2,A_0), (A_2,A_0,A_0), (A_0,A_3,A_4), (A_3,A_4,A_2)\), \((A_4,A_2,A_0), (A_0,A_0,A_3) \}\), and its transition matrix \(A_5\) is \(A_5=B=\left[ \begin{array}{ccccccccc} 1&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1 \\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0 \\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0 \\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0\\ 1&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1\\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0\\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0\\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0\\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0&{}\ 0 \end{array} \right] \).

  6. 6.

    The determinative block system \(\fancyscript{A}_6\) is \(\fancyscript{A}_6=\{(A_0,A_1,A_2),(A_1,A_2,A_0), (A_2,A_0,A_0), (A_0,A_0,A_0), (A_0,A_0,A_1),(A_4,A_3,A_2), (A_3,A_2,A_0), (A_0,A_0,A_4), (A_0,A_4,A_3) \}\), and its transition matrix \(A_6\) is \(A_6=\left[ \begin{array}{ccccccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ \end{array} \right] \).

  7. 7.

    The determinative block system \(\fancyscript{A}_7\) is \(\fancyscript{A}_7=\{(A_0,A_1,A_2), (A_1,A_2,A_0), (A_2,A_0,A_0), (A_0,A_0,A_0), (A_0,A_0,A_1), (A_1,A_0,A_2), (A_0,A_2,A_0), (A_0,A_1,A_0) \}\), and its transition matrix \(A_7\) is \(A_7=\left[ \begin{array}{cccccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 \\ \end{array} \right] \).

  8. 8.

    The determinative block system \(\fancyscript{A}_8\) is \(\fancyscript{A}_8=\{(A_3,A_4,A_2), (A_4,A_2,A_0),(A_2,A_0,A_0), (A_0,A_0,A_3), (A_0,A_3,A_4), (A_4,A_3,A_2), (A_3,A_2,A_0), (A_0,A_0,A_4), (A_0,A_4,A_3), (A_0,A_0,A_0) \}\), and its transition matrix \(A_8\) is \(A_8=\left[ \begin{array}{cccccccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ \end{array} \right] \).

  9. 9.

    The determinative block system \(\fancyscript{A}_9\) is \(\fancyscript{A}_9=\{(A_3,A_4,A_2), (A_4,A_2,A_0),(A_2,A_0,A_0), (A_0,A_0,A_3), (A_0,A_3,A_4), (A_1,A_0,A_2), (A_0,A_2,A_0), (A_0,A_0,A_1), (A_0,A_1,A_0), (A_0,A_0,A_0) \}\), and its transition matrix \(A_9\) is \(A_9=\left[ \begin{array}{cccccccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ \end{array} \right] \).

  10. 10.

    The determinative block system \(\fancyscript{A}_{10}\) is \(\fancyscript{A}_{10}=\{(A_4,A_3,A_2), (A_3,A_2,A_0),(A_2,A_0,A_0), (A_0,A_0,A_4), (A_0,A_4,A_3),(A_1,A_0,A_2),(A_0,A_2,A_0), (A_0,A_0,A_1), (A_0,A_1,A_0), (A_0,A_0,A_0) \}\), and its transition matrix \(A_{10}\) is \(A_{10}=\left[ \begin{array}{cccccccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ \end{array} \right] \).

  11. 11.

    The determinative block system \(\fancyscript{A}_{11}\) is \(\fancyscript{A}_{11}=\{(A_0,A_1,A_2), (A_1,A_2,A_0),(A_2,A_0,A_0), (A_0,A_0,A_0), (A_0,A_0,A_1), (A_3,A_4,A_2), (A_4,A_2,A_0), (A_0,A_0,A_3), (A_0,A_3,A_4), (A_4,A_3,A_2), (A_3,A_2,A_0), (A_0,A_0,A_4), (A_0,A_4,A_3)\}\), and its transition matrix \(A_{11}\) is \(A_{11}=\left[ \begin{array}{ccccccccccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ \end{array} \right] \).

  12. 12.

    The determinative block system \(\fancyscript{A}_{12}\) is \(\fancyscript{A}_{12}=\{(A_0,A_1,A_2), (A_1,A_2,A_0),(A_2,A_0,A_0), (A_0,A_0,A_0), (A_0,A_0,A_1), (A_3,A_4,A_2), (A_4,A_2,A_0), (A_0,A_0,A_3)\), \( (A_0,A_3,A_4), (A_1,A_0,A_2), (A_0,A_2,A_0), (A_0,A_1,A_0)\}\), and its transition matrix \(A_{12}\) is \(A_{12}=\left[ \begin{array}{cccccccccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 \\ \end{array} \right] \).

  13. 13.

    The determinative block system \(\fancyscript{A}_{13}\) is \(\fancyscript{A}_{13}=\{(A_0,A_1,A_2), (A_1,A_2,A_0),(A_2,A_0,A_0), (A_0,A_0,A_0), (A_0,A_0,A_1),(A_4,A_3,A_2), (A_3,A_2,A_0), (A_0,A_0,A_4), (A_0,A_4,A_3) , (A_1,A_0,A_2), (A_0,A_2,A_0), (A_0,A_1,A_0)\}\), and its transition matrix \(A_{13}\) is \(A_{13}=\left[ \begin{array}{cccccccccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 \\ \end{array} \right] \).

  14. 14.

    The determinative block system \(\fancyscript{A}_{14}\) is \(\fancyscript{A}_{14}=\{(A_3,A_4,A_2), (A_4,A_2,A_0),(A_2,A_0,A_0), (A_0,A_0,A_3), (A_0,A_3,A_4), (A_4,A_3,A_2), (A_3,A_2,A_0), (A_0,A_0,A_4), (A_0,A_4,A_3), (A_1,A_0,A_2), (A_0,A_2,A_0) (A_0,A_0,A_1), (A_0,A_1,A_0), (A_0,A_0,A_0)\}\), and its transition matrix \(A_{14}\) is \(A_{14}=\left[ \begin{array}{cccccccccccccc} 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 0 \\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 0 &{}\ 0 &{}\ 1 &{}\ 0 &{}\ 1 \\ \end{array} \right] \).

  15. 15.

    The determinative block system \(\fancyscript{A}_{15}\) is \(\fancyscript{A}_{15}=\{(A_0,A_1,A_2), (A_1,A_2,A_0),(A_2,A_0,A_0), (A_0,A_0,A_0), (A_0,A_0,A_1), (A_3,A_4,A_2), (A_4,A_2,A_0), (A_0,A_0,A_3), (A_0,A_3,A_4), (A_4,A_3,A_2), (A_3,A_2,A_0), (A_0,A_0,A_4), (A_0,A_4,A_3), (A_1,A_0,A_2), (A_0,A_2,A_0), (A_0,A_1,A_0)\}\), and its transition matrix \(A_{15}\) is \(A_{15}\!=\!\left[ \begin{array}{cccccccccccccccc} 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 \\ 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 \\ 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 \\ 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 0 &{}\, 1 &{}\, 0 &{}\, 0 \\ \end{array} \right] \).

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Jin, W., Chen, F. A novel construction of chaotic dynamics base gliders in diffusion rule B2/S7. Nonlinear Dyn 78, 779–788 (2014). https://doi.org/10.1007/s11071-014-1476-0

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