Fractal dimension of the controlled Julia sets of the output duopoly competing evolution model

Volume 16, Issue 3, pp 461-471 http://dx.doi.org/10.22436/jmcs.016.03.16

Publication Date: September 15, 2016 Submission Date: March 22, 2016

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Authors

Zhaoqing Li - School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, China. Yongping Zhang - School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, China. Jian Liu - School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China.

Abstract

The output duopoly competing evolution model has an integral role in the study of the economic phenomenon. In this paper, the basic methods of Julia sets is applied to this model. At first, Julia set of this model is introduced. Then, two different control methods are taken to control Julia set: one is the step hysteresis control method and the other is the optimal function control. Meanwhile box-counting dimensions of the controlled Julia set under these methods are computed to depict the complexity of Julia sets and the system. The simulation results show the efficacy of these methods.

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Keywords

• The output duopoly competing evolution model
• Julia set
• step hysteresis control method
• optimal function control fractal dimension.

MSC

•  26A33
•  26E70

References

• [1] H. N. Agiza, A. S. Hegazi, A. A. Elsadany , The dynamics of Bowley's model with bounded rationality , Chaos Solitons Fractals, 12 (2001), 1705-1717.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] L. Budinski-Petković, I. Lončarević, Z. M. Jakšić, S. B. Vrhovac, Fractal properties of financial markets , Phys. A, 410 (2014), 43-53.
  • View Article
  • Google Scholar


• [3] J. G. Du, T. Huang, Z. Sheng , Analysis of decision-making in economic chaos control, Nonlinear Anal. Real World Appl., 10 (2009), 2493-2501.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] K. Falconer, Fractal geometry , Mathematical foundations and applications, John Wiley & Sons, Ltd., Chichester (1990)


• [5] Y. Lin , Fractal and its application in the securities market, Econ. Probl., 8 (2001), 44-46.


• [6] P. Liu , Spatial fractal control and chaotic synchronization in complex dynamical system, Shandong Univ., (2012), 22-25.


• [7] F. Lu, Fractal theory and its application in economic management , Financ. Manag. Inst., 3 (2011), 91-93.


• [8] Z. Peng, C. Li, H. Li , The research of enterprise and fractal management model in Era of knowledge economy , Business Studies, 255 (2002), 33-35.


• [9] D. Preiss, Nondiffierentiable functions, (Czech) Pokroky Mat. Fyz. Astronom., 28 (1983), 148-154.


• [10] J. Shu, D. Tan, J. Wu , The exploration of fractal structure in China's stock market , J. Southwest Jiaotong Univ., 38 (2004), 212-215.


• [11] A. S. Soliman , Fractals in nonlinear economic dynamic systems , Chaos Solitons Fractals, 7 (1996), 247-256.
  • View Article
  • Google Scholar


• [12] M. de Sousa Vieira, A. J. Lichtenberg , Controlling chaos using nonlinear feedback with delay , Phys. Rev. E, 54 (1996), 1200-1207.
  • View Article
  • Google Scholar


• [13] C. Wang, R. Chu, J. Ma , Controlling a chaotic resonator by means of dynamic track control, Complexity, 21 (2015), 370-378.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] F. Zhang, C. Mu, G. Zhang, D. Lin, Dynamics of two classes of Lorenz-type chaotic systems, Complexity, 21 (2015), 363-369.
  • View Article
  • MathSciNet
  • Google Scholar


• [15] M. Zhang, Z. Wei , The Application of the fractal theory in describing the regional transportation economy coordination, China soft sci., 1 (1996), 112-123.


• [16] C. Zhang, F. Wu, L. Zhao , The application of the fractal theory in economics, Statistics and Decision, 1 (2009), 158-159.


• [17] X. H. Zhu, N. Patel, W. Zuo, X. C. Yang, Fractal analysis applied to spatial structure of chinas vegetation , Chinese Geographical Sci., 16 (2006), 48-55.
  • View Article
  • Google Scholar