Spatio-temporal chaos in duopoly games

Volume 10, Issue 7, pp 3784--3791 http://dx.doi.org/10.22436/jnsa.010.07.33

Publication Date: July 26, 2017 Submission Date: April 28, 2017

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Authors

Risong Li - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China. Yu Zhao - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China. Tianxiu Lu - Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China. - Artificial Intelligence Key Laboratory of Sichuan Province, Zigong, Sichuan, 643000, P. R. China. Ru Jiang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China. Hongqing Wang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China. Haihua Liang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China.

Abstract

Suppose that \(G\) and \(H\) are two given closed subintervals of \(R\), and that \(q : G \rightarrow H\) and \(p : H \rightarrow G\) are continuous maps. Let \(\Gamma (s, t) = (p(t), q(s))\) be a Cournot map over the space \(G \times H\). In this paper, we study spatio-temporal chaos of such a Cournot map. In particular, it is shown that if \(p\) and \(q\) are onto maps, then the following are equivalent: 1) \(\Gamma\) is spatio-temporally chaotic; 2) \(\Gamma^2\mid_{\Lambda_1}\) is spatio-temporally chaotic; 3) \(\Gamma^2\mid_{\Lambda_2}\) is spatio-temporally chaotic; 4) \(\Gamma\mid_{\Lambda_1\cup\Lambda_2}\) is spatio-temporally chaotic. Moreover, it is proved that if \(p\) and \(q\) are onto maps, then \(p \circ q\) is spatio-temporally chaotic if and only if so is \(q \circ p\). Also, we give two examples which show that for the above results, it is necessary to assume that \(p\) and \(q\) are onto maps.

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Keywords

• Spatio-temporal chaos
• Li-Yorke sensitivity
• duopoly game.

MSC

•  37D45
•  54H20
•  37B40
•  26A18
•  28D20

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