Journal of Applied Mathematics

Chaos for Discrete Dynamical System

Lidong Wang, Heng Liu, and Yuelin Gao

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We prove that a dynamical system is chaotic in the sense of Martelli and Wiggins, when it is a transitive distributively chaotic in a sequence. Then, we give a sufficient condition for the dynamical system to be chaotic in the strong sense of Li-Yorke. We also prove that a dynamical system is distributively chaotic in a sequence, when it is chaotic in the strong sense of Li-Yorke.

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J. Appl. Math., Volume 2013 (2013), Article ID 212036, 4 pages.

First available in Project Euclid: 14 March 2014

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Wang, Lidong; Liu, Heng; Gao, Yuelin. Chaos for Discrete Dynamical System. J. Appl. Math. 2013 (2013), Article ID 212036, 4 pages. doi:10.1155/2013/212036.

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  • T. Y. Li and J. A. Yorke, “Period three implies chaos,” The Ameri-can Mathematical Monthly, vol. 82, no. 10, pp. 985–992, 1975.
  • L. D. Wang, G. F. Huang, and S. M. Huan, “Distributional chaos in a sequence,” Nonlinear Analysis, vol. 67, no. 7, pp. 2131–2136, 2007.
  • P. Oprocha, “Distributional chaos revisited,” Transactions of the American Mathematical Society, vol. 361, no. 9, pp. 4901–4925, 2009.
  • M. Martelli, Introduction to Discrete Dynamical Systems and Chaos, Discrete Mathematics and Optimization, John Wiley & Sons, New York, NY, USA, 1999.
  • M. Martelli, M. Dang, and T. Seph, “Defining chaos,” Mathematics Magazine, vol. 71, no. 2, pp. 112–122, 1998.
  • L. Liu and S. Zhao, “Martelli's Chaos in inverse limit dynamical systems and hyperspace dynamical systems,” Results in Mathematics, vol. 63, no. 1-2, pp. 195–207, 2013.
  • S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2, Springer, New York, NY, USA, 1990.
  • L. Wang, G. Liao, and Y. Yang, “Recurrent point set of the shift on $\Sigma $ and strong chaos,” Annales Polonici Mathematici, vol. 78, no. 2, pp. 123–130, 2002.
  • R. S. Yang, “Distribution chaos in a sequence and topologically mixing,” Acta Mathematica Sinica, vol. 45, no. 4, pp. 752–758, 2002 (Chinese).
  • G. Liao, L. Wang, and Y. Zhang, “Transitivity, mixing and chaos for a class of set-valued mappings,” Science in China A, vol. 49, no. 1, pp. 1–8, 2006.