Narrow Results

The dynamical and fractal behaviors of the complex perturbed rational maps are discussed in this paper. And the optimal control function method is taken on the Julia set of this system. In this control method, infinity is regarded as a fixed point to be controlled. By substituting the driving item for an item in the optimal control function, synchronization of Julia sets of two such different systems is also studied.

The forced Brusselator model is investigated from the fractal viewpoint. A Julia set of the discrete version of the Brusselator model is introduced and control of the Julia set is presented by using feedback control. In order to discuss the relations of two different Julia sets, a coupled item is designed to realize the synchronization of two Julia sets with different parameters, which provides a method to discuss the relation and the changing of two different Julia sets, one Julia set can be changed to be the other. Numerical simulations are used to verify the effectiveness of these methods.

It is of great significance to study the three-dimensional financial system model based on the discrete fractional-order theory. In this paper, the Julia set of the three-dimensional discrete fractional-order financial model is identified to show its fractal characteristics. The sizes of the Julia sets need to be changed in some situations, so it is necessary to achieve control of the Julia sets. In combination with the characteristics of the model, two different controllers based on the fixed point are designed, and the control of the three-dimensional Julia sets is realized by adding the controllers into the model in different ways. Finally, the simulation graphs show that the controllers can effectively control the fractal behaviors.

Lotka–Volterra population competition model plays an important role in mathematical models. In this paper, Julia set of the competition model is introduced by use of the ideas and methods of Julia set in fractal geometry. Then feedback control is taken on the Julia set of the model. And synchronization of two different Julia sets of the model with different parameters is discussed, which makes one Julia set change to be another. The simulation results show the efficacy of these methods.

It is of crucial significance to study the infectious disease phenomenon by using the SIRS model and thoughts of Julia set. In this paper, Julia set of the discrete version of the SIRS model is established to analyze the fractal dynamics of this model. Then, controller is designed to change the Julia set. Furthermore, the box-counting dimensions of the controlled Julia sets by selecting different appropriate parameters are computed to show the complexity of the model. Finally, a nonlinear coupling method is introduced to synchronize the Julia sets with different parameters of the same system. Simulation results show the efficacy of these methods.