Research on Synchronization of Fractal Behaviors in 2-D Logistic Map

In this paper, the fractal behaviors in 2-D Logistic map are discussed. First, the definition and some properties of fractal set in 2-D Logistic map are given. Moreover, several synchronous definitions between different 2-D Logistic maps are introduced in this paper, such as complete synchronization, coupled synchronization, and adaptive synchronization. The synchronization of the fractal set between different Logistic maps is accomplished by synchronizing their iterative tracks. The simulations illustrate the effectiveness and correctness of these methods.


I. INTRODUCTION
Logistic map is a classic model for studying the behaviors of complex systems such as dynamic systems, chaos, and fractals. It is mainly used in epidemiology, such as it can be used to explore the risk factors of a disease. And it can also be used to simulate the growth behavior of biological populations. Therefore, Logistic map is also called "insect population model", its expression is 1 (1 )   is a tunable parameter. There are lots of achievements about the chaotic behaviors of Logistic map, which relate to chaotic control and applications [1], chaotic cryptography [2]- [7], the image encryption [8] and many other research areas.
Zhang [9] achieved the control of the Julia set of the complex Henon system by using the gradient control and the auxiliary reference feedback control, respectively. Liu [10] achieved the control and synchronization of the Julia set in coupled map lattice by using the gradient control and the optimal control. Zhang [11] achieved the control and synchronization of the Mandelbrot set using the feedback control. Wang [12,13] discussed the bifurcation and fractal behaviors of the coupled Logistic maps.
The fractal behaviors in 2-D Logistic map, with abundant nonlinear characteristics such as typical selfsimilarity and initial value sensitivity, can be widely used in the research of secure communication. Therefore, the synchronous research of fractal behaviors in 2-D Logistic map is particularly important. However, there are fewer results on the synchronization of fractal behaviors in 2-D Logistic map. In this paper, Julia set in 2-D Logistic map is taken as an example, to discuss the synchronous control under different definitions in different application scenarios.
The outline of the paper is as follows. In the section of Julia set in 2-D Logistic map, the definitions of Julia set in Logistic map and the fixed points are given. In the section of Synchronization, several definitions of synchronization of the Julia set in 2-D Logistic map are given. In the section of Simulations, the synchronizations of two different Julia set under these definitions are achieved. The example shows the feasibility of these methods. Finally, the conclusions are given in the section of Conclusions.

II. JULIA SET IN 2-D LOGISTIC MAP
In this section, some necessary knowledge about fractional theory and Julia set is reviewed briefly. More relevant contents can be found in monographs [14]- [18] and references contained therein.
In this paper, in order to facilitate the discussion of the fractal set in 2-D Logistic map, its discrete form is taken as follows,  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.  [14] of Julia set, the filled Julia set of System (1) is the set of the points ( , ) xy whose trajectories are limited, that is the Julia set of System (1) is the boundary of the filled-in Julia set K , which is denoted by J , that is JK = . The Julia set J of System (1) has the following properties [15]. (1) J is nonempty and bounded; (2) J is fully invariant, According to the definition of the Julia set in Logistic map, the structure of its Julia set is closely related to the iterative trajectory of System (1). Therefore, the synchronous control of the Julia set in Logistic map can be achieved by controlling its iterative trajectory.

III. DEFINITIONS OF DIFFERENT TYPES OF SYNCHRONIZATION
In order to elicit the definition of synchronization, two different 2-D Logistic maps with controller are taken, 11 (1 ) , where   . To achieve the synchronization of Julia set in Systems (2) and (3), Some definitions [19] of synchronization of Julia set between two different 2-D Logistic maps will be reviewed.  (2), it can be referred to as a drive system, whose Julia set is denoted as 2 o J , and System  (2) and (3) achieve complete synchronization.
From Equation (7), we have Obviously, the equivalent expression of Equation (9) is ( ) And the unknown parameter  can be identified.
In the light of Julia sets definition, the spatial Julia sets are four-dimensional graphics. In order to plot the graphics of the spatial Julia sets, we fix one of four real coordinates and use a three-dimensional graphic to simulate the spatial Julia sets. In this paper, the same real and imaginary parts of y in System (1) are taken to make a graph.

Case 1: Coupled Synchronization
Combining Expressions (13) and (14), for the given  and  , there exist  Figure 1 illustrates the original Julia sets of Systems (2) and (3) (2) and (3). Therefore, Systems (2) becomes For the same initial value 00 xw = , then we have ( ) Combining Expressions (16) and (17), for the given  and  , there exist 11 0 nn xw ++ −→ when 1 k + →  , that is k →. Therefore, when k →, the Julia sets of Systems (15) and (3) achieve completed synchronization. increasing of k , and it becomes the same with the Julia set of System (3) as k → . In other words, the synchronization of the Julia set is accomplished, which approves the above conclusion.

Case 3: Adaptive Synchronization
The adaptive controller    Figure 6 shows the estimation of the unknown parameter  .

V. CONCLUSION
In this paper, some definitions of synchronization of Julia sets between different 2-D Logistic maps are discussed. Moreover, we discuss the adaptive synchronization between two different 2-D Logistic maps with the unknown parameters. The parameter estimator is designed, and the identification of unknown parameters is completed based on the known conditions. Some examples are taken to certificate the effectiveness of these three types of synchronization. These control methods and their theories are successfully applied to other aspects of fractal theory, which can help us to explain the corresponding complicated phenomena better.