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 On account of the complexity of the fractal set, the traditional Euclidean dimension is no longer applicable and the measurement method of fractal dimension is required.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.

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 the early 20th century, the fundamental theory of chaos and fractal was proposed. The theory explains the unity of determinacy and randomness and the unity of order and disorder. It is considered to be the third major revolution of science after the theory of relativity and quantum mechanics [

Fractal theory, first proposed in the 1970s, comes from the study of nonlinear science. Its primary research object is the geometric form of nature and nonlinear system, which is complex but has some kind of self similarity and regularity. In 1977, Mandelbrot, a professor of mathematics of the Harvard University, published the landmark work

Nowadays, with the emergence of some new mathematical tools and methods, especially the combination of the study of fractal theory and computer, the theory has been developed rapidly. In addition, researchers not only constantly establish and improve the theory of fractals, but also apply it in various fields, such as the diffusion processes and chemical kinetics in crowded media, the protein structure and complex vascular branches in biomedicine, dynamical system and hydromechanics in physics, and landforms evolution and earthquake monitoring [

Considering the complexity of fractal sets, traditional Euclidean geometry dimension cannot accurately depict their geometric forms. Mathematicians propose many definitions of noninteger dimension and use different names to distinguish them. For example, 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 [

Brusselator model [

Notably, in the nonlinear system, the Julia set of the system is an important nonlinear feature. According to the objective requirement, we often need restrict the size of the nonlinear attractive domain. And sometimes it is required that the system possess different or similar behavior and performance in compliance with the actual requirements of technical problems. As a result, how to effectively control the Julia set is particularly critical.

Based on the Julia set of Brusselator model, feedback control method, optimal control method, and gradient control method [

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. He noticed that functions on the complex plane as simple as

Take

Let

In fractal theory, fractal dimension is one of the most elemental concepts. At present, there are many definitions of fractal dimensions, including Hausdorff dimension, box-counting dimension, similarity dimension. In all kinds of definitions of fractal dimensions, Hausdorff dimension is the basis of the fractal theory. It can even be considered the theoretical basis of the fractal geometry. However, Hausdorff dimension is just suitable for the theoretical analysis of fractal theory, and there is only a small class of fairly solid mathematical regular fractal graphics that can be calculated for their Hausdorff dimension. It is hard to calculate the fractal dimension which is proposed in the practical applications. Therefore, people propose the concept of box-counting dimension. Its popularity is largely due to its relative ease of mathematical calculation and empirical estimation. In fact, in practical applications, the dimension is generally referred to as box-counting dimension. The precise definition of box-counting dimension is as follows.

Let

If these are equal we refer to the common value as the

Brusselator model is one of the most fundamental models in nonlinear systems, and the dynamic equations are as follows:

Brusselator equations were first discovered by A. Turing in 1952 [

Set

The research results demonstrate that, coupled with a piecewise constant value control function, we can control the size of the limit cycle of (

As was mentioned above, considering the stability of the fixed points of system (

Take

Let

Write

And the characteristic equation is

When the modulus of the eigenvalues

By Theorem

For example, take

Six simulation diagrams are chosen corresponding to the values of

In the same control, six simulation diagrams are chosen corresponding to the values of

In feedback control method, the contraction of the left and the lower parts is faster than the right and the upper parts when the interval of control parameter

From the perspective of the change of box-counting dimensions, with the absolute values of

The original Julia set of the system.

The change of the Julia sets of the controlled system when

The change of box-counting dimensions of the Julia sets of the controlled system when

The change of the Julia sets of the controlled system when

The change of box-counting dimensions of the Julia sets of the controlled system when

Take

Let

Write

And the characteristic equation is

When the modulus of the eigenvalue

By Theorem

For example, we take the values of system parameters

Six simulation diagrams are chosen corresponding to the values of

In optimal control method, the effective controlled interval of

From the perspective of the change of box-counting dimensions, box-counting dimensions of the Julia sets generally show a monotonic decreasing trend with the absolute values of

The change of the Julia sets of the controlled system when

The change of box-counting dimensions of the Julia sets of the controlled system when

Take

Let

Write

And the characteristic equation is

When the modulus of the eigenvalue

By Theorem

For example, we take the values of system parameters

Six simulation diagrams are chosen corresponding to the values of

In the same control, six simulation diagrams are chosen corresponding to the values of

In gradient control method, we consider two groups of interval of the parameter

From the perspective of the change of box-counting dimensions, with the absolute values of

The change of the Julia sets of the controlled system when

The change of box-counting dimensions of the Julia sets of the controlled system when

The change of the Julia sets of the controlled system when

The change of box-counting dimensions of the Julia sets of the controlled system when

Fractal theory is a hot topic in the research of nonlinear science. Describing the change of chemical elements in the chemical reaction process, Brusselator model, an important class of reaction diffusion equations, is significant in the study of chaotic and fractal behaviors of nonlinear differential equations. In technological applications, it is often required that the behavior and performance of the system can be controlled effectively.

In this paper, feedback control method, optimal control method, and gradient control method are taken to control the Julia set of Brusselator model, respectively, and the box-counting dimensions of the Julia set are calculated. In each control method, when the absolute value of control parameter

The authors declare that they have no competing interests.

This work is supported by the National Natural Science Foundation of China (nos. 61403231 and 11501328) and China Postdoctoral Science Foundation (no. 2016M592188).