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The fractional Potts model on diamond-like hierarchical lattices is introduced in this manuscript, which is a fractional rational system in the complex plane. Then, the fractal dynamics of this model is discussed from the fractal viewpoint. Julia set of the fractional Potts model is given, and control items of this fractional model are designed to control the Julia set. To associate two different Julia sets of the fractional model with different parameters and fractional orders, nonlinear coupling items are taken to make one Julia set change to another. The simulations are provided to illustrate the efficacy of these methods.

The model of a diamond hierarchical lattice in statistical physics shows that there is an important relation between the limit point set of zero points of the partition function and the Julia set of a class of rational functions. Julia set is one of the important sets in fractal theory which is used extensively in many fields [

Hamiltonian for

Fractional calculus is a generalization of the ordinary differential and integral to an arbitrary order. The fractional dynamical systems are related with the past status and can reflect the situation of the system more realistically [

In the practical applications, the behaviors depicted by the Julia set need to show different forms or one behavior under control changes to be another. Therefore, it is necessary to study the control of Julia sets [

For convenience, we give some necessary definitions and results on the discrete fractional calculus.

(see [

(see [

(see [

The domains of equations (

The integer form can be obtained for

From Theorem

Consequently, the numerical equation can be proposed:

In particular, if

The fractional system (

Julia set is one of the important sets in fractal theory, which is generated by the iteration of the integer order system. We will give the definition of the Julia set for the discrete fractional system (

Let

To give the simulations of the Julia set of system (

Step 1: take the parameters in the equation.

Step 2: loop through the items about

Step 3: reuse a loop about

Step 4: take a boundary about the Julia set; these pairs of points are iterating to form a graph, and the boundary of the image is the Julia set.

Take some different values of

Figure

Julia sets of the discrete fractional system (

Julia set control is a hot topic, and many control methods are introduced in recent years [

Let

The solutions of (

Introduce the control item into system (

Since 1 is the fixed point for any

Figures

Changing of Julia sets of the discrete fractional system (

Changing of Julia sets of the discrete fractional system (

Figures

In fact, the control item can be added to the other locations in system (

Though the shapes of the controlled Julia sets are different, the control results are similar under these three ways (

Synchronization of nonlinear systems is an interesting topic and is applied extensively in mechanics, communication, and so on [

Consider a system with the same form as (

Figure

Synchronization of Julia sets of (

Figures

Figures

Figures

The fractional Potts model on diamond-like hierarchical lattices is introduced in this manuscript, which is a generalization of the Potts model with the integer order. By using the fractal theory, the Julia set of the discrete fractional of the model is given. Then, control of the Julia set is discussed by use of the fixed point 1. Some original values not present in the Julia set can be seen in the Julia set by the control. Julia sets are determined and not associated with each other when the values of the system parameters are different. Coupling items are designed to make one Julia set change to another. The simulations illustrate the efficacy of these methods.

The figure data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China-Shandong Joint Fund (no. U1806203), Natural Science Foundation of Shandong Province (no. ZR2019MA051), Fundamental Research Funds for the Central Universities (no. 2019ZRJC005), and National Natural Science Foundation of China key fund (no. 61533011).