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Discrete Dynamics in Nature and Society
Volume 2016, Article ID 8234108, 13 pages
Research Article

Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model

School of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, China

Received 21 September 2016; Accepted 7 November 2016

Academic Editor: Cengiz Çinar

Copyright © 2016 Yuqian Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Fractal theory is a branch of nonlinear scientific research, and its research object is the irregular geometric form in nature. On account of the complexity of the fractal set, the traditional Euclidean dimension is no longer applicable and the measurement method of fractal dimension is required. In the numerous fractal dimension definitions, box-counting dimension is taken to characterize the complexity of Julia set since the calculation of box-counting dimension is relatively achievable. In this paper, the Julia set of Brusselator model which is a class of reaction diffusion equations from the viewpoint of fractal dynamics is discussed, and the control of the Julia set is researched by feedback control method, optimal control method, and gradient control method, respectively. Meanwhile, we calculate the box-counting dimension of the Julia set of controlled Brusselator model in each control method, which is used to describe the complexity of the controlled Julia set and the system. Ultimately we demonstrate the effectiveness of each control method.