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Journal of Mathematical Chemistry

, Volume 53, Issue 5, pp 1220–1226 | Cite as

Topological entropy and \({\fancyscript{P}}\)-chaos of a coupled lattice system with non-zero coupling constant related with Belusov–Zhabotinskii reaction

  • Jinxing Liu
  • Tianxiu Lu
  • Risong Li
Original Paper

Abstract

This paper focuses on the chaotic properties of the following systems stated by Kaneko (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction:
$$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f(x_{n}^{m})+ \frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f(x_{n+1}^{m})\right] , \end{aligned}$$
where \(m\) is discrete time index, \(n\) is lattice side index with system size \(L\), \(\varepsilon \in (0, 1]\) is coupling constant and \(f\) is a continuous selfmap of \([0, 1]\). It is shown that for every continuous selfmap \(f\) of the interval \([0, 1]\), the topological entropy of such a coupled lattice system is not less than the topological entropy of the map \(f\), and that for every continuous selfmap of the interval \([0, 1]\) with positive topological entropy, such a system is \({\fancyscript{P}}\)-chaotic, where \({\fancyscript{P}}\) denotes one of the three properties: Li–Yorke chaos, distributional chaos, \(\omega \)-chaos. These results extend the ones of Wu and Zhu (J Math Chem 50:1304–1308, 2012), (J Math Chem 50:2439–2445, 2012) and Li et al. (J Math Chem 51:1712–1719, 2013).

Keywords

Coupled map lattice \({\fancyscript{P}}\)-chaos Topological entropy 

Mathematics Subject Classification

54H20 58F03 47A16 

Notes

Acknowledgments

This paper was partially supported by the Scientific Research Fund of Sichuan Provincial Education Department (No. 14ZB0007).

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of FinanceSouthwestern University of Finance and EconomicsChengduPeople’s Republic of China
  2. 2.School of MathematicsYibin UniversityYibinPeople’s Republic of China
  3. 3.Department of MathematicsSichuan University of Science and EngineeringZigongPeople’s Republic of China
  4. 4.School of ScienceGuangdong Ocean UniversityZhanjiangPeople’s Republic of China

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