Kato's chaos and P-chaos of a coupled lattice system given by Garcia Guirao and Lampart which is related with Belusov-Zhabotinskii reaction

Document Type: Research Paper

Author

Guangdong Ocean University

10.22052/ijmc.2020.148532.1390

Abstract

In this article, we further consider the above system. In particular, we give a sufficient condition under which the above system is Kato chaotic for $\eta=0$ and a necessary condition for the above system to be Kato chaotic for $\eta=0$. Moreover, it is deduced that for $\eta=0$, if $\Theta$ is P-chaotic then so is this system, where a continuous map $\Theta$ from a compact metric space $Z$ to itself is said to be P-chaotic if it has the pseudo-orbit-tracing property and the closure of the set of all periodic points for $\Theta$ is the space $Z$. Also, an example and three open problems are presented.

Keywords


 

  1. T. Y. Li and J. A. Yorke, Period three implies chaos, Am. Math. Mon. 82 (10) (1975) 985-992.
  2. L. S. Block and W. A. Coppel, Dynamics in One Dimension, Springer Monographs in Mathematics, Springer, Berlin, 1992.
  3. R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin/ Cumings, Menlo Park, CA, 1986.
  4. J. R. Chazottes and B. Fern Andez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Physics (Berlin Heidelberg New York Springer), Vol. 671, 2005.
  5. J. L. García Guirao and M. Lampart, Positive entropy of a coupled lattice system related with Belusov-Zhabotinskii reaction, J. Math. Chem. 48 (2010) 66-71.
  6. K. Kaneko, Globally coupled chaos violates law of large numbers, Phys. Rev. Lett. 65 (1990) 1391-1394.
  7. J. L. García Guirao and M. Lampart, Chaos of a coupled lattice system related with Belusov-Zhabotinskii reaction, J. Math. Chem. 48 (2010) 159-164.
  8. M. Kohmoto and Y. Oono, Discrete model of chemical turbulence, Phys. Rev. Lett. 55 (1985) 2927-2931.
  9. J. L. Hudson, M. Hart and D. Marinko, An experimental study of multiplex peak periodic and nonperiodic oscilations in the Belusov-Zhabotinskii reaction, J. Chem. Phys. 71 (1979) 1601-1606.
  10. K. Hirakawa, Y. Oono and H. Yamakazi, Experimental study on chemical turbulence II, J. Phys. Soc. Jap. 46 (1979) 721-728.
  11. J. L. Hudson, K. R. Graziani, R. A. Schmitz, Experimental evidence of chaotic states in the Belusov-Zhabotinskii reaction, J. Chem. Phys. 67 (1977) 3040-3044.
  12. D. Ruelle and F. Takens, On the natural of turbulence, Comm. Math. Phys. 20 (1971) 167-92.
  13. H. Kato, Everywhere chaotic homeomorphisms on manifields and k-dimensional Menger manifolds, Topol. Appl. 72 (1996) 1-17.
  14. R. Gu, Kato’s chaos in set-valued discrete systems, Chaos, Solitons & Fractals 31 (2007) 765-771.
  15. G. L. Forti, Various notions of chaos for discrete dynamical systems, A brief survey, Aequationes Math. 70 (2005) 1-13.
  16. X. Wu and P. Zhu, On sensitive dependence of continuous interval mappings (In Chinese), J. Systems Sci. Math. Sci. 32 (2012) 215-225.
  17. T. Arai and N. Chinen, P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy, Topol. Appl. 154 (2007) 1254-1262.  
  18. X. Wu, P. Oprocha and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity 29 (2016) 1942-1972.
  19. F. Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci. 1 (2) (2016) 391-404.