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Comment on “A note on the principal measure and distributional \((p, q)\)-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction”

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The Original Article was published on 08 March 2013

Abstract

In García Guirao and Lampart (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this note, we give an example which shows that the proofs of Theorems 3.1 and 3.2 in [J Math Chem 51:1410–1417, 2013] are incorrect, and two open problems.

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References

  1. T.Y. Li, J.A. Yorke, Period three implies chaos. Am. Math. Mon. 82(10), 985–992 (1975)

    Article  Google Scholar 

  2. L.S. Block, W.A. Coppel, Dynamics in One Dimension, Springer Monographs in Mathematics (Springer, Berlin, 1992)

    Google Scholar 

  3. R.L. Devaney, An Introduction to Chaotics Dynamical Systems (Benjamin/Cummings, Menlo Park, CA, 1986)

    Google Scholar 

  4. J.R. Chazottes, B. FernSndez, Dynamics of coupled map lattices and of related spatially extended systems. Lecturer Notes in Physics, vol 671 (2005)

  5. J.L. García Guirao, M. Lampart, Chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction. J. Math. Chem. 48, 159–164 (2010)

    Article  Google Scholar 

  6. K. Kaneko, Globally coupled chaos violates law of large numbers. Phys. Rev. Lett. 65, 1391–1394 (1990)

    Article  Google Scholar 

  7. X.X. Wu, P.Y. Zhu, Li–Yorke chaos in a coupled lattice system related with Belusov–Zhabotinskii reaction. J. Math. Chem. 50, 1304–1308 (2012)

    Article  CAS  Google Scholar 

  8. B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Am. Math. Soc. 344, 737–754 (1994)

    Article  Google Scholar 

  9. P. Oprocha, P. Wilczyński, Shift spaces and distributional chaos. Chaos Solitons Fractals 31, 347–355 (2007)

    Article  Google Scholar 

  10. J. Smítal, M. Stefánková, Distributional chaos for triangular maps. Chaos Solitons Fractals 21, 1125–1128 (2004)

    Article  Google Scholar 

  11. R. Pikula, On some notions of chaos in dimension zero. Colloq. Math. 107, 167–177 (2007)

    Article  Google Scholar 

  12. X.X. Wu, P.Y. Zhu, A minimal DC1 system. Topol. Appl. 159, 150–152 (2012)

    Article  Google Scholar 

  13. X.X. Wu, P.Y. Zhu, The principal measure and distributional \((p, q)\)-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction. J. Math. Chem. 50, 2439–2445 (2012)

    Article  CAS  Google Scholar 

  14. R. Li, X. Zhou, Y. Zhao, C. Huang, A note on the principal measure and distributional \((p, q)\)-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction. J. Math. Chem. 51, 1410–1417 (2013)

    Article  CAS  Google Scholar 

  15. D.L. Yuan, J.C. Xiong, Densities of trajectory approximation time sets. Sci. Sin. Math. 40(11), 1097–1114 (2010). (in Chinese)

    Google Scholar 

  16. B. Schweizer, A. Sklar, J. Smítal, Distributional (and other) chaos and its measurement. Real Anal. Exch. 21, 495–524 (2001)

    Google Scholar 

  17. M. Kohmoto, Y. Oono, Discrete model of chemical turbulence. Phys. Rev. Lett. 55, 2927–2931 (1985)

    Article  Google Scholar 

  18. J.L. Hudson, M. Hart, D. Marinko, An experimental study of multiplex peak periodic and nonperiodic oscilations in the Belusov–Zhabotinskii reaction. J. Chem. Phys. 71, 1601–1606 (1979)

    Article  CAS  Google Scholar 

  19. K. Hirakawa, Y. Oono, H. Yamakazi, Experimental study on chemical turbulence II. J. Phys. Soc. Jpn. 46, 721–728 (1979)

    Article  Google Scholar 

  20. J.L. Hudson, K.R. Graziani, R.A. Schmitz, Experimental evidence of chaotic states in the Belusov–Zhabotinskii reaction. J. Chem. Phys. 67, 3040–3044 (1977)

    Article  Google Scholar 

  21. G. Chen, S.T. Liu, On spatial periodic orbits and spatial chaos. Int. J. Bifurc. Chaos 13, 935–941 (2003)

    Article  Google Scholar 

Download references

Acknowledgments

This research was supported by the NSF of Guangdong Province (Grant 10452408801004217), the Key Scientific and Technological Research Project of Science and Technology Department of Zhanjiang City (Grant 2010C3112005), the Science and Technology Promotion Special of Ocean and Fisheries of Guangdong Province (A201008A05), and Guangdong science and technology plan projects (Grant 2009B030803014).

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Correspondence to Risong Li.

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This comment refers to the article available at (doi:10.1007/s10910-013-0155-6).

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Li, R. Comment on “A note on the principal measure and distributional \((p, q)\)-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction”. J Math Chem 52, 775–780 (2014). https://doi.org/10.1007/s10910-014-0310-8

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  • DOI: https://doi.org/10.1007/s10910-014-0310-8

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