Abstract
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 On account of the complexity of the fractal set, the traditional Euclidean dimension is no longer applicable and the measurement method of fractal dimension is required.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.
Highlights
In order to recognize the essence of some extremely sophisticated phenomena, researchers attempt to figure out the regularity and unity which exist behind these phenomena so that they can control and predict them better
In 1918, Julia Gaston, a famous French mathematician, discovered an important fractal set in fractal theory, when he studied the iteration of complex functions, which was named Julia set
When the control parameter k is in the interval from 0.16 to 0.37 and −0.9 to −0.4, the monotonic change of box-counting dimensions is obvious, which indicates the effectiveness of this control method on the Julia set of Brusselator model
Summary
In order to recognize the essence of some extremely sophisticated phenomena, researchers attempt to figure out the regularity and unity which exist behind these phenomena so that they can control and predict them better. Hausdorff dimension which was proposed by the German mathematician Hausdorff in 1919 has a rigorous mathematical definition It is established on the basis of Hausdorff measure and can define most fractal sets, so it is easier to deal with in mathematics [20]. Brusselator model [25,26,27] is a kind of reaction diffusion equations which describe the change of chemical elements in the process of chemical reaction [28] It is significant in the study of the chaos and fractal behavior of nonlinear differential equations. Researchers have studied Brusselator model from different aspects and proven some properties of it [29,30,31,32,33] These studies of the model have contributed much to the development of nonlinear mathematics. The box-counting dimension of the Julia set of controlled Brusselator model is calculated in each control method to describe the complexity of the Julia set of the system
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