Remark on topological entropy and \({\mathscr {P}}\) -chaos of a coupled latti

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• 作者：Risong Li ; Jianjun Wang ; Tianxiu Lu ; Ru Jiang
• 关键词：Coupled map lattice ; \({\mathscr {P}}\) ; chaos ; Topological entropy
• 刊名：Journal of Mathematical Chemistry
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：54
• 期：5
• 页码：1110-1116
• 全文大小：386 KB
• 参考文献：1.R.L. Adler, A.G. Konheim, M.H. McAndrew, Topological entropy. Trans. Amer. Math. Soc. 309–319 (1965)
  2.L.S. Block, W.A. Coppel, Dynamics in One Dimension, Springer Monographs in Mathematics (Springer, Berlin, 1992)
  3.R. Bowen, Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153, 401–414 (1971)CrossRef
  4.R.A. Dana, L. Montrucchio, Dynamical complexity in duopoly games. J. Econ. Theory 40, 40–56 (1986)CrossRef
  5.R.L. Devaney, An Introduction to Chaotics Dynamical Systems (Benjamin/Cummings, Menlo Park, CA, 1986)
  6.E.I. Dinaburg, A connection between various entropy characterizations of dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 35, 324–366 (1971)
  7.G.L. Forti, L. Paganoni, J. Smítal, Strange triangular maps of the square. Bull. Aust. Math. Soc. 51, 395–415 (1995)CrossRef
  8.J.L. García Guirao, M. Lampart, Positive entropy of a coupled lattice system related with Belusov-Zhabotinskii reaction. J. Math. Chem. 48, 66–71 (2010)CrossRef
  9.J.L. García Guirao, M. Lampart, Chaos of a coupled lattice system related with Belusov-Zhabotinskii reaction. J. Math. Chem. 48, 159–164 (2010)CrossRef
  10.K. Kaneko, Globally coupled chaos violates law of large numbers. Phys. Rev. Lett. 65, 1391–1394 (1990)CrossRef
  11.K. Kaneko, H.F. Willeboordse, Bifurcations and spatial chaos in an open ow model. Phys. Rev. Lett. 73, 533–536 (1994)CrossRef
  12.M. Kohmoto, Y. Oono, Discrete model of chemical turbulence. Phys. Rev. Lett. 55, 2927–2931 (1985)CrossRef
  13.R. Li, F. Huang, Y. Zhao, Z. Chen, C. Huang, The principal measure and distributional \((p, q)\) -chaos of a coupled lattice system with coupling constant \(\varepsilon =1\) related with Belusov-Zhabotinskii reaction. J. Math. Chem. 51, 1712–1719 (2013)CrossRef
  14.S. Li, \(\omega \) -chaos and topological entropy. Trans. Amer. Math. Soc. 339, 243–249 (1993)
  15.T.Y. Li, J.A. Yorke, Period three implies chaos. Amer. Math. Mon. 82, 985–992 (1975)CrossRef
  16.J. Liu, T. Lu, R. Li, Topological entropy and \({\fancyscript {P}}\) -chaos of a coupled lattice system with non-zero coupling constant related with Belusov-Zhabotinskii reaction. J. Math. Chem. 53, 1220–1226 (2015)CrossRef
  17.P. Oprocha, Invariant scrambled sets and distributional chaos. Dyn. Syst. 24, 31–43 (2009)CrossRef
  18.P. Oprocha, P. Wilczyński, Shift spaces and distributional chaos. Chaos Solitons Fractals 31, 347–355 (2007)CrossRef
  19.R. Pikula, On some notions of chaos in dimension zero. Colloq. Math. 107, 167–177 (2007)CrossRef
  20.T. Puu, Chaos in duopoly pricing. Chaos, Solitions and Fractals 1, 573–581 (1991)CrossRef
  21.B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc. 344, 737–754 (1994)CrossRef
  22.A.N. Sharkovskii, Coexistence of cycles of a continuous mapping of the line into itself. Ukrainian Math. J. 16, 61–71 (1964)
  23.J. Smítal, M. Stefánková, Distributional chaos for triangular maps. Chaos Solitons Fractals 21, 1125–1128 (2004)CrossRef
  24.B. VanderPool, Forced oscilations in a circuit with nonlinear resistence. London, Edinburgh and Dublin. Philos. Mag. 3, 109–123 (1927)
  25.X. Wu, P. Zhu, Li–Yorke chaos in a coupled lattice system related with Belusov- Zhabotinskii reaction. J. Math. Chem. 50, 1304–1308 (2012)CrossRef
  26.X. Wu, P. 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)CrossRef
  27.X. Wu, P. Zhu, A minimal DC1 system. Topol. Appl. 159, 150–152 (2012)CrossRef
  
• 作者单位：Risong Li (1)
  Jianjun Wang (2)
  Tianxiu Lu (3)
  Ru Jiang (1)
  
  1. School of Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People’s Republic of China
  2. Department of Mathematics, Sichuan Agricultural University, Yaan, Sichuan, 625014, People’s Republic of China
  3. Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, People’s Republic of China
  
• 刊物类别：Chemistry and Materials Science
• 刊物主题：Chemistry
  Physical Chemistry
  Theoretical and Computational Chemistry
  Mathematical Applications in Chemistry
  
• 出版者：Springer Netherlands
• ISSN：1572-8897

文摘

In this paper we study some chaotic properties of the following systems which is posed by Kaneko in (Phys Rev Lett, 65: 1391-1394, 1990) and is related to the Belusov-Zhabotinskii reaction: $$\begin{aligned} u_{b}^{a+1}=(1-\alpha )r(u_{b}^{a})+ \frac{1}{2}\alpha [r(u_{b-1}^{a})+r(u_{b+1}^{a})], \end{aligned}$$where a is discrete time index, b is lattice side index with system size T, \(\alpha \in [0, 1]\) is coupling constant and r is a continuous selfmap on \(J=[0, 1]\). It is proven that for each continuous selfmap r on J, the topological entropy of such a coupled lattice system with \(\alpha =0\) is not less than the topological entropy of r, and that for each continuous selfmap on J with positive topological entropy, the above system with \(\alpha =0\) is \({\mathscr {P}}\)-chaotic, where \({\mathscr {P}}\) is one of the three properties: Li–Yorke chaos, distributional chaos, \(\omega \)-chaos. Moreover, we deduce that for each continuous selfmap r on J and any \(\alpha \in [0, 1]\), if r is \(\omega \)-chaotic, then so is the above system. Keywords Coupled map lattice \({\mathscr {P}}\)-chaos Topological entropy Mathematics Subject Classification 54H20 58F03 47A16 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (27) References1.R.L. Adler, A.G. Konheim, M.H. McAndrew, Topological entropy. Trans. Amer. Math. Soc. 309–319 (1965)2.L.S. Block, W.A. Coppel, Dynamics in One Dimension, Springer Monographs in Mathematics (Springer, Berlin, 1992)3.R. Bowen, Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153, 401–414 (1971)CrossRef4.R.A. Dana, L. Montrucchio, Dynamical complexity in duopoly games. J. Econ. Theory 40, 40–56 (1986)CrossRef5.R.L. Devaney, An Introduction to Chaotics Dynamical Systems (Benjamin/Cummings, Menlo Park, CA, 1986)6.E.I. Dinaburg, A connection between various entropy characterizations of dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 35, 324–366 (1971)7.G.L. Forti, L. Paganoni, J. Smítal, Strange triangular maps of the square. Bull. Aust. Math. Soc. 51, 395–415 (1995)CrossRef8.J.L. García Guirao, M. Lampart, Positive entropy of a coupled lattice system related with Belusov-Zhabotinskii reaction. J. Math. Chem. 48, 66–71 (2010)CrossRef9.J.L. García Guirao, M. Lampart, Chaos of a coupled lattice system related with Belusov-Zhabotinskii reaction. J. Math. Chem. 48, 159–164 (2010)CrossRef10.K. Kaneko, Globally coupled chaos violates law of large numbers. Phys. Rev. Lett. 65, 1391–1394 (1990)CrossRef11.K. Kaneko, H.F. Willeboordse, Bifurcations and spatial chaos in an open ow model. Phys. Rev. Lett. 73, 533–536 (1994)CrossRef12.M. Kohmoto, Y. Oono, Discrete model of chemical turbulence. Phys. Rev. Lett. 55, 2927–2931 (1985)CrossRef13.R. Li, F. Huang, Y. Zhao, Z. Chen, C. Huang, The principal measure and distributional \((p, q)\)-chaos of a coupled lattice system with coupling constant \(\varepsilon =1\) related with Belusov-Zhabotinskii reaction. J. Math. Chem. 51, 1712–1719 (2013)CrossRef14.S. Li, \(\omega \)-chaos and topological entropy. Trans. Amer. Math. Soc. 339, 243–249 (1993)15.T.Y. Li, J.A. Yorke, Period three implies chaos. Amer. Math. Mon. 82, 985–992 (1975)CrossRef16.J. Liu, T. Lu, R. Li, Topological entropy and \({\fancyscript {P}}\)-chaos of a coupled lattice system with non-zero coupling constant related with Belusov-Zhabotinskii reaction. J. Math. Chem. 53, 1220–1226 (2015)CrossRef17.P. Oprocha, Invariant scrambled sets and distributional chaos. Dyn. Syst. 24, 31–43 (2009)CrossRef18.P. Oprocha, P. Wilczyński, Shift spaces and distributional chaos. Chaos Solitons Fractals 31, 347–355 (2007)CrossRef19.R. Pikula, On some notions of chaos in dimension zero. Colloq. Math. 107, 167–177 (2007)CrossRef20.T. Puu, Chaos in duopoly pricing. Chaos, Solitions and Fractals 1, 573–581 (1991)CrossRef21.B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc. 344, 737–754 (1994)CrossRef22.A.N. Sharkovskii, Coexistence of cycles of a continuous mapping of the line into itself. Ukrainian Math. J. 16, 61–71 (1964)23.J. Smítal, M. Stefánková, Distributional chaos for triangular maps. Chaos Solitons Fractals 21, 1125–1128 (2004)CrossRef24.B. VanderPool, Forced oscilations in a circuit with nonlinear resistence. London, Edinburgh and Dublin. Philos. Mag. 3, 109–123 (1927)25.X. Wu, P. Zhu, Li–Yorke chaos in a coupled lattice system related with Belusov- Zhabotinskii reaction. J. Math. Chem. 50, 1304–1308 (2012)CrossRef26.X. Wu, P. 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)CrossRef27.X. Wu, P. Zhu, A minimal DC1 system. Topol. Appl. 159, 150–152 (2012)CrossRef About this Article Title Remark on topological entropy and \({\mathscr {P}}\) -chaos of a coupled lattice system with non-zero coupling constant related with Belusov-Zhabotinskii reaction Journal Journal of Mathematical Chemistry Volume 54, Issue 5 , pp 1110-1116 Cover Date2016-05 DOI 10.1007/s10910-016-0609-8 Print ISSN 0259-9791 Online ISSN 1572-8897 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Physical Chemistry Theoretical and Computational Chemistry Math. Applications in Chemistry Keywords Coupled map lattice $${\mathscr {P}}$$ P -chaos Topological entropy 54H20 58F03 47A16 Industry Sectors IT & Software Oil, Gas & Geosciences Telecommunications Authors Risong Li (1) Jianjun Wang (2) Tianxiu Lu (3) Ru Jiang (1) Author Affiliations 1. School of Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People’s Republic of China 2. Department of Mathematics, Sichuan Agricultural University, Yaan, Sichuan, 625014, People’s Republic of China 3. Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, People’s Republic of China Continue reading... To view the rest of this content please follow the download PDF link above.