Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model
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 On account of the complexity of the fractal set, the traditional Euclidean dimension is no longer applicable and the measurement method of fractal dimension is required.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...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2016/8234108 |
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| _version_ | 1832546095812575232 |
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| author | Yuqian Deng Xiuxiong Liu Yongping Zhang |
| author_facet | Yuqian Deng Xiuxiong Liu Yongping Zhang |
| author_sort | Yuqian Deng |
| collection | DOAJ |
| description | 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 On account of the complexity of the fractal set, the traditional Euclidean dimension is no longer applicable and the measurement method of fractal dimension is required.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. |
| format | Article |
| id | doaj-art-e670e8cab2d04d77b5931d881fea51d7 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-e670e8cab2d04d77b5931d881fea51d72025-02-03T07:23:53ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2016-01-01201610.1155/2016/82341088234108Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator ModelYuqian Deng0Xiuxiong Liu1Yongping Zhang2School of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, ChinaSchool of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, ChinaSchool of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, ChinaFractal 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.http://dx.doi.org/10.1155/2016/8234108 |
| spellingShingle | Yuqian Deng Xiuxiong Liu Yongping Zhang Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model Discrete Dynamics in Nature and Society |
| title | Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model |
| title_full | Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model |
| title_fullStr | Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model |
| title_full_unstemmed | Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model |
| title_short | Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model |
| title_sort | fractal dimension analysis of the julia sets of controlled brusselator model |
| url | http://dx.doi.org/10.1155/2016/8234108 |
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