Finite-Time Synchronization of Fractional-Order Chaotic Systems with Different Structures under Stochastic Disturbances

This paper studies the finite-time synchronization of fractional-order chaotic systems with different structures under parameter disturbance and external disturbance. We put forward a fractional-order controller that can achieve the finite-time synchronization of any-order fractional-order chaotic systems under stochastic disturbances. This controller has good robustness and an-ti-interference performance. With the concept of the finite-time stability theory given, some judgment criterions for the synchronization of fraction-al-order chaotic systems are proved. This method can not only make the error systems have a faster convergence rate but also can be implemented in engineering easily. The numerical simulations of two specific examples dem-onstrate the effectiveness of the method. At the same time, the synchronised time of finite-time synchronization is shorter and faster than the complete synchronization and the time can be adjusted according to the parameters in the controller.


Introduction
Although the development history of fractional and integer calculus is not much different, it turns out that fractional calculus is of great significance to express the model we are studying. This is because it not only has the memory function but also more accurately describes the fractional order dynamic models. Of course, fractional calculus has many other advantages [1] [2] [3]. It's because of these benefits that the theories of fractional calculus are widely used in physics [4], engineering [2], chemistry [5] and other subjects. In recent decades, these fields including information processing [6], secure communication [7], thermal systems [8], robot control and problems [9] have been discovered that many physical processes exhibit fractional order dynamic behavior.
Chaos has achieved tremendous and far-reaching development in many fields after discovering the first chaotic attractor by Lorenz. At present, the research on chaos control and synchronization has spread across many disciplines. In the past 20 years, many methods of chaos control have appeared, such as OGY control [10], drive-response control [11], pulse control [12], sliding mode control [13], active control [14], Lyapunov direct method [15], etc. Taking into account above methods, some scholars have considered the synchronization problem for a class of fractional-order chaotic systems [16] [17] [18] while others only concentrate on the specific chaotic systems [19] [20] [21]. And many scholars have solved the synchronization problem of chaotic systems of any order [22] [23].
Currently, the main research direction is to make the system reach synchronization in infinite time, that is, to consider the asymptotically stable of systems.
However, facing the actual situation, we may need a specified time, that is, finite-time synchronization. It not only has a faster convergence rate but also stronger robustness [24]. Keyong S et al. used the finite-time synchronization theories to define a nonlinear controller which realizes that the synchronization of four-dimensional fractional order hyperchaotic system [25]. Lingdong Z et al.
adopted the finite-time stability theory to accomplish the synchronization of hyperchaotic Lorenz systems [26]. In literature [27], a robust non-singular terminal sliding mode controller is proposed for synchronizing two different input nonlinear uncertain chaotic systems. However, these results are aimed at specific chaotic systems. What we need more is a controller that can control all fractional-order chaotic systems. What's more, in real life, chaotic systems do not exist in society in isolation. Of course, there are some disturbances. For example, the signal transmission caused by random fluctuations is a disturbed process. Disturbances can be divided into parameter disturbance [28], external disturbance [29] and internal disturbance [30]. Their presence will make the system unstable and difficult to control. Therefore, some scholars began to consider the impact of disturbance. Literatures [31] [32] [33] have solved the synchronization of fractional-order chaotic systems under random disturbance, but they only considered one of the above three disturbances. If they can consider multiple disturbances, it has more practical significance. Although literatures [34] [35] [36] consider the case of multiple disturbances, the object of their research is the integer order chaotic system.
In response to this situation, we are going to consider the synchronization of fractional-order chaotic systems under two kinds of disturbances, namely para-

Definitions and Lemmas of Fractional Derivative
Next, we will introduce the Riemann-Liouville (R-L) derivative and the Caputo derivative. When the order α is a negative real number and a positive integer, they are equivalent. The R-L definition is more suitable for theoretical analysis and can simplify the calculation of fractional order derivatives. The definition of Caputa is more suitable for modern engineering and makes the Laplace transformation more concise.
The mathematical expression of Caputo derivative with order α is given as  has continuous first derivative, then where ( ) 0,1 α ∈ and Q is an arbitrary n order positive definite matrix.

Stability Theories of Fractional Order System
Considering that most of the things around us are nonlinear, we write the fractional order nonlinear system to be:   (4) is called finite-time stable. The stabilization time of the system (4) has the following form Corollary 1. It follows from Lemma 1 and Theorem 2 that the system (4) must first satisfy the criterions of the Mittag-Leffler stability. From the conditions (2) and (3) in Theorem 2, we get (8) Hence, the system (4) is called finite-time stable if it satisfies (8) and the criterions of the Mittag-Leffler stability.

Sufficients Condition for Finite-Time Synchronization
In this chapter, we believe that a small disturbance can make a great change in the orbit of the chaotic system. Therefore, it is reasonable to treat them as bounded. This will also make our theory easier to understand. With the help of Lyapulov function, we obtain our conclusions successfully.
The fractional-order drive-response system with parametric disturbance and external disturbance is demonstrated as follows. The drive system is: x f x d x (9) Journal of Computer and Communications The response system is: (10) where ( ) We suppose the state error among the driving system with response system as (10) to get the error system: For the sake of ensuring our conclusion more realistic, we need to make the following assumptions. (12) where 0 M > , ⋅ represents the 2-norm of matrix.
Theorem 3. When the assumptions 1 -3 are all satisfied, the systems (9) and (10) are stable for a finite time with the following controller: x e e (15) where 1 m L p q ≥ + + , 2 m l M ≥ + and 0 ≠ e .
Proof. Take the Lyapulov function as The fractional order derivative of the Lyapulov function is   (17) Finally, is obtained. Therefore, we must be able to find 3 is held, we can obtain that the systems (9) and (10) where ( )

Numerical Simulation
We exemplify a pair of three-dimensional and four-dimensional fractional chaotic systems to affirm the availability of our method. The results indicate that the error variables of the system quickly stabilize in a finite time, and the synchronization time is able to adjust via 1 2 , m m . Among them, the influence of 1 m on the synchronization time is small, and we can ignore it.
We use the MATLAB software to obtain the state variable trajectory of the systems (19) and (20) obtaining the following results  According to (19) and (20), we get the following matrices. The parameter matrices are 35 35 0 10 10 0 7 28 0 , The parameter interference matrices are  (23) (24) It follows from the assumptions 1 -3 and Theorem 3 that the following calculation results are obtained.
Hence, it follows from Assumption 3 and Theorem 3 that we get We get the controller as ( ) Under the control of the above controller, the synchronised time satisfies 4.21 T ≤ seconds. Numerical simulations show that the (19) and (20)     , , , Let the fractional order hyperchaotic Liu chaotic system [40] under stochastic disturbances and controller be the response system , , , When we select the initial value as ( ) 11, 2,1, 1 − and the order 0.99 α = , the driving system (26) appears chaotic attractors which are presented in Figure 6. We realize that the trajectories about the state variables of fractional order hyperchaotic Lorenz and Liu systems are not synchronized with time without any control. Next, we will verify the effectiveness of our controller.
We use the MATLAB software to obtain the state variable trajectory of the systems (26) and (27)  According to (26) and (27) The parameter interference matrices are   , (30) The external disturbances are ( ) ( ) It follows from the assumptions 1 -3 and Theorem 3 that the following calculation results are obtained. x x x − − .

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Hence, it follows from Assumption 3 and Theorem 3 that we get We get the controller as ( ) Under the control of above controller, the synchronised time satisfies 6.47 T ≤ seconds. Numerical simulations show that the (26) and (27) achieve the finite-time synchronization revealing in Figures 8-10. Comparing Figures 8-10, it's worth noting that the synchronization time gradually decreases as the value of 2 m increases. We can also say that the synchronised time of finite-time synchronization is shorter and faster than the complete synchronization in infinite time, and it can be adjusted according to the 2 m . Journal of Computer and Communications

Conclusion
In

Authors Contributions
WQ PAN proposed the main the idea and prepared the manuscript initially. TZ LI gave the numerical simulation of this paper.

Availability of Data and Material
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that they have no competing interests.