Advertisement

Journal of Mathematical Chemistry

, Volume 52, Issue 3, pp 775–780 | Cite as

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

  • Risong Li
Commentary
  • 123 Downloads

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.

Keywords

Coupled map lattice Distributional (p, q)-chaos Principal measure Devaney’s chaos Chaos in the sense of Li–Yorke Tent map 

Mathematics Subject Classification

54H20 37B40 37D45 

Notes

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).

References

  1. 1.
    T.Y. Li, J.A. Yorke, Period three implies chaos. Am. Math. Mon. 82(10), 985–992 (1975)CrossRefGoogle Scholar
  2. 2.
    L.S. Block, W.A. Coppel, Dynamics in One Dimension, Springer Monographs in Mathematics (Springer, Berlin, 1992)Google Scholar
  3. 3.
    R.L. Devaney, An Introduction to Chaotics Dynamical Systems (Benjamin/Cummings, Menlo Park, CA, 1986)Google Scholar
  4. 4.
    J.R. Chazottes, B. FernSndez, Dynamics of coupled map lattices and of related spatially extended systems. Lecturer Notes in Physics, vol 671 (2005)Google Scholar
  5. 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)CrossRefGoogle Scholar
  6. 6.
    K. Kaneko, Globally coupled chaos violates law of large numbers. Phys. Rev. Lett. 65, 1391–1394 (1990)CrossRefGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 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)CrossRefGoogle Scholar
  9. 9.
    P. Oprocha, P. Wilczyński, Shift spaces and distributional chaos. Chaos Solitons Fractals 31, 347–355 (2007)CrossRefGoogle Scholar
  10. 10.
    J. Smítal, M. Stefánková, Distributional chaos for triangular maps. Chaos Solitons Fractals 21, 1125–1128 (2004)CrossRefGoogle Scholar
  11. 11.
    R. Pikula, On some notions of chaos in dimension zero. Colloq. Math. 107, 167–177 (2007)CrossRefGoogle Scholar
  12. 12.
    X.X. Wu, P.Y. Zhu, A minimal DC1 system. Topol. Appl. 159, 150–152 (2012)CrossRefGoogle Scholar
  13. 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)CrossRefGoogle Scholar
  14. 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)CrossRefGoogle Scholar
  15. 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. 16.
    B. Schweizer, A. Sklar, J. Smítal, Distributional (and other) chaos and its measurement. Real Anal. Exch. 21, 495–524 (2001)Google Scholar
  17. 17.
    M. Kohmoto, Y. Oono, Discrete model of chemical turbulence. Phys. Rev. Lett. 55, 2927–2931 (1985)CrossRefGoogle Scholar
  18. 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)CrossRefGoogle Scholar
  19. 19.
    K. Hirakawa, Y. Oono, H. Yamakazi, Experimental study on chemical turbulence II. J. Phys. Soc. Jpn. 46, 721–728 (1979)CrossRefGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. 21.
    G. Chen, S.T. Liu, On spatial periodic orbits and spatial chaos. Int. J. Bifurc. Chaos 13, 935–941 (2003)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.School of ScienceGuangdong Ocean UniversityZhanjiang People’s Republic of China

Personalised recommendations