Comparison of Times of Synchronization of Chaotic Burke-Shaw Attractor with Active Control and Integer and Optimum Fractional-order Pecaro Carroll Method

– Many studies have been introduced in the literature showing that two identical chaotic systems can be synchronized with different initial conditions. Secure data communication applications have also been made using synchronization methods. In the study, synchronization times of two popular synchronization methods are compared, which is an important issue for communication. Among the synchronization methods, active control, integer, and fractional-order Pecaro Carroll (P-C) method was used to synchronize the Burke-Shaw chaotic attractor. The experimental results showed that the P-C method with optimum fractional-order is synchronized in 2.35 times shorter time than the active control method. This shows that the P-C method using fractional-order creates less delay in synchronization and is more convenient to use in secure communication applications.


Introduction
Since chaotic systems are the most complex nonlinear steady-state behavior, they were considered unsuitable for synchronization. However, Pecora and Carroll showed that two identical chaotic systems with different initial conditions can synchronize and introduced the P-C method [1,2]. After the P-C method was introduced, many synchronization methods were introduced. For the synchronization of two identical systems with different initial conditions, adaptive control [3], linear and nonlinear feedback control [4,5], active control [6,7], and passive control [8] have been proposed.
It has become a popular topic with the spread of synchronization methods and has been used in security applications. Using the synchronization of two identical chaotic systems, secure communication and information hiding studies have been introduced [9][10][11]. Liao and Lin [3] dealt with unknown system parameters and adaptive control and synchronization problems of Lorenz systems. Huang and et al. [4] synchronized two identical chaotic systems with a nonlinear control method based on Lyapunov stability theory. They tried their proposed method by applying it to two identical Lü systems and two different chaotic systems. Park [5] proposed a new nonlinear controller based on the Lyapunov stability theory by addressing the chaos synchronization problem of a chaotic system. The proposed controller ensures that the states of the controlled chaotic dependent system asymptotically synchronize the states of the host system. Yassen [6] presented chaos synchronization between two different chaotic systems using active control. Uçar et al. [7] investigated the synchronization of combined chaotic systems through active control. Synchronization is done in the slave-master program. The controller ensures that the states of the chaotic dependent system are exponentially synchronized with the state of the host system. Wang and Liu [8] investigated passive control synchronization. In their work, they designed a passive controller and performed synchronization of two identical combined chaotic systems with different initial conditions. Pehlivan et al. [9] used the P-C method to synchronize two identical chaotic systems with different initial conditions and implement chaotic encrypted information signal transfer. Akgül et al. [10] performed a data hiding method with a three-dimensional chaotic system without an equilibrium point. In the study, a new data hiding method based on chaos that hides an image of a different color to color images is proposed. Through the proposed method, the data is hidden in bit spaces inside the image with the help of a chaotic random number generator (RNG). Cheng and cheng [11] tried to develop a robust synchronization scheme for two different chaotic systems exposed to limited noise. The synchronization controller they proposed is embedded in a secure communication scheme that is not only resistant to channel noise but can also include the noise as part of the encryption key, thus increasing the key security. Ojo et al. [12] examined synchronization methods in their study and presented their advantages and disadvantages compared to each other. Khan and Chaudhary [13] studied the hybrid projective combination synchronization scheme between identical chaotic generalized Lotka-Volterra three types of biological systems using active control design. Kocamaz and Uyaroğlu [14] presented the chaos synchronization and anti-synchronization of the Single Machine Infinite Bus (SMIB) power system to the Duffing oscillator using the active control method. Matouk [15] synchronized the autonomous Duffing-Van der Pol system modified with integer-order and fractional-order P-C method. Also, the synchronization between two different fractional-order chaotic systems is studied and the fractionalorder Lu system is controlled as the fractional-ordered Chen system. In the work of Zhu et al. [16] based on the stability theorem of linear fractional systems, a necessary condition is given to control the chaos synchronization of disproportionate ordered fractional systems. Wang et al. [17] proposed the synchronization of a fractional time delay financial system using a new type-2 fuzzy active control method. Sahin et al. [18] proposed a new four-dimensional (4D) chaotic system comprising an active flux controlled memristor characterized by a uniform continuous cubic nonlinearity. Yu et al. [19] proposed a new 5D hyperchaotic four-bladed memristive system (HFWMS) by developing a quadratic nonlinear flux-controlled memristor to the 4-dimensional hyperchaotic system as the feedback term. Alkhedhairi [20] recently introduced a new financial model based on a non-smooth fractional-order Caputo-Fabrizio derivative. Chaos synchronization between the two master / slave fractional financial models has been obtained based on the adaptive control theory. Carbajal-Gomez et al. [21] proposed a current mode segmented linear (PWL) function based on CMOS cells that allow programmable generation of 2-7 shift chaotic attractors. Also, they showed that it is suitable for secure communication by providing synchronization of 2-7 slide towers in a master-slave topology with generalized Hamilton forms and observer approach. Bhardwaj and Das [22] investigated the synchronization problem of two-three types of models with the Beddington-DeAngelis type functional response using the master-dependent scheme. Based on the selected intrinsic growth parameters that characterize the complex dynamics of the master and dependent system, active controller functions that cause general asymptotic stability of synchronization errors for cases I and II, respectively, are obtained. Emiroğlu and Uyaroğlu [23] have synchronized Burke-Shaw's chaotic attraction with active control.
The most important issue in security applications created using the synchronization method is the synchronization time of the two chaotic systems. The shorter the synchronization time, the better the communication quality in secure communication applications. This is due to the short delay created by the synchronization times. When the delay in communication is short, the communication quality increases.
In this study, the Burke-Shaw chaotic attractor is introduced and the Burke-Shaw chaotic system, which has two different starting points, has been synchronized with both the active control method and the P-C method with integer and optimum fractional-order value. Each synchronization method is modeled and implemented in Matlab ™ environment. It is shown with error graphs that two systems are synchronized in each synchronization system. In the study, synchronization times of synchronization methods were compared. The experimental results showed that the optimum fractional-order value P-C method synchronizes in 2.3 (time unit) less than the active control method and 4.3 (time unit) compared to the integer-order P-C method. This shows that the P-C method using fractional-order creates less delay in synchronization and is more convenient to use in secure communication applications. In the second part of this study, Matlab-Simulink modeling is given by introducing the Burke-Shaw chaotic attractor. In the third chapter, synchronization with the active control method is explained and synchronization graphics are given. In the fourth chapter, synchronization with fractional-order P-C method is explained, synchronization graphics are given and synchronization times of all synchronization methods are compared. Results are given in the last section.

Burke-Shaw Chaotic Attractor
Burke-Shaw chaotic attractor was created by Bill Burke and Robert Shaw derived from the Lorenz chaotic equation [24]. The equation set for the Burke-Shaw chaotic attractor is given in (1). In (1) has x, y, and z state variables and s and V system parameters. As seen in the equation in (1), it is seen that the Burke-Shaw chaotic attractor is a 3rd-degree chaotic system. The Burke-Shaw chaotic attractor has been implemented in the Matlab-Simulink environment and the Matlab-Simulink model is given in Fig. 1  Lyapunov exponentials of the Burke-Shaw chaotic attractor are calculated with (2) and the Lyapunov graph is plotted in Matlab environment and is given in Fig. 4.

Synchronization with Active Control
Two identical Burke-Shaw chaotic attractors with different initial conditions are synchronized by the active control method. In (3), the driver system equation of the active control method is given. The response system that will respond to the driver system is given in (4).

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In (4), u1(t), u2(t), and u3(t) are control functions that provide synchronization with active control between the driver and the responder system and are given in (5). The v1(t), v2(t) and v3(t) given in (5)  x s x y u (t) Driver and responder in (3) and (4) are defined as the error between the two systems. To find the synchronization errors, the driver system is removed from the answering system, and the error dynamic system in (6) In active control, control inputs v1(t), v2(t), and v3(t) control the error system in (6), converting the error to zero, which indicates that synchronization has been achieved. The control inputs v1(t), v2(t), and v3(t) are selected as shown in the matrix in (8).
In (8), the B matrix is a constant matrix, and all eigenvalues must be negative for the error system in (7) to be asymptotically stable. All eigenvalues of matrix B given in (8) were chosen to be negative. In the last case, the error dynamic system is given in (9). The Lyapunov function suitable for the error cases e1(t), e2(t), and e3(t) in the dynamic system of the error obtained in (9) is given in (10). When the derivative of the Lyapunov function in (10) is taken, the equation in (11)  dV eee dt     (11) Since the V in (11) is negative, the driver (3) and the responder (4) systems are asymptotically stable, which is proof that the two systems are synchronized. The systems were run by taking different initial conditions T T x y z   respectively for driver and receiver systems implemented in Matlab environment. Fig. 5 shows the x-y-z time series of the driver and responder systems before active synchronization and the synchronization error signals between the two systems. When the x-y-z time series in Fig. 5.a, b and c are examined, it is seen that the two systems are not synchronized and have different values at the same time.
When the error signals showing the difference of x-y-z state variables between the driver and receiver systems in Fig.  5.d are examined, it shows that the driver and receiver systems are not synchronized, all state variables generate values throughout the entire time series and do not converge to zero. In Fig. 6, after active synchronization, the x-y-z time series of the driver and responder systems and the synchronization error signals between the two systems are given. When the x-y-z time series in Fig. 6.a, b and c are examined, it is seen that the two systems are synchronized and all state variables of the two systems are at the same values after 4 (time unit) time. When the error signals showing the difference of x-y-z state variables between the driver and receiver systems in Fig. 6.d are examined, the convergence of all state variables to zero after 4 (time unit) time shows that the driver and receiver systems are synchronized.

Synchronization with Fractional-order P-C Method
Pecora and Carroll arbitrarily divide an original chaotic system they discussed in their work into two separate parts and named them driver and responder subsystems. They showed that by creating the same responsive subsystem in the receiver module, chaotic synchronization can be achieved if this subsystem is driven with the driver part of the original system. In other words, they have shown both theoretically and experimentally that the chaotic signal generated in the receiver module will converge to the chaotic signal coming from the original system [1]. Fig. 4 shows the operating system of the P-C synchronization method. As seen in Fig. 4, the x, y, and z state variables directly stimulate the responder subsystem. The responder system is divided into two subsystems. In the responder system, variable x for the first subsystem is warned from the driver system, while variable y for the second subsystem is warned from the first subsystem. Fractional-order applied Burke-Shaw chaotic attractor is synchronized with the P-C method. Fig. 7 The block diagram of P-C synchronization [1].
In the P-C method, the equation of the fractional-order Burke-Shaw chaotic attractor driver system given in Fig. 7 is given in (12).

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The first order ( 2 y & and 2 z &) stable first sub responder system of the P-C method is given in (13).
Likewise, a second-order ( 3 x && and 3 z && ) stable second sub responder system is given in (14).
In the P-C synchronization realized with integer-order in the Matlab environment, the driver (12) and receiver systems (13,14) were taken with the same initial conditions as the active control, and the systems were operated. In Fig.  8, while all conditions are the same, synchronization error signals between the two systems with the x-y-z time series of the driver and responder systems after P-C synchronization with integer-order are given. When the x-y-z time series in  Fig. 8.a, b and c are examined, it is seen that the two systems are synchronized and all state variables of the two systems are at the same values after 6 (time unit) time. When the error signals showing the difference of x-y-z state variables between the driver and receiver systems in Fig. 8.d are examined, the convergence of all state variables to zero after 6 (time unit) time shows that the driver and receiver systems are synchronized. In the P-C synchronization implemented using fractional-order in Matlab environment, the driver (12) and receiver systems (13,14) were re-operated under the same initial conditions as the active control. The Burke-shaw fractional order chaotic system was modelled with nid function, one of Ninteger fractional control toolbox for Matlab tools [25]. The Burke-Shaw chaotic P-C synchronization system was tested with different fractional-order values in the range of 0<q<1 and the optimum fractional-order value was tried to be found. Here, the condition that ensures optimum fractional-order is the condition of the chaotic system to be synchronized as soon as possible. Fig. 9 shows the synchronization error graphs of the chaotic system in P-C synchronization, which is implemented using the shortest-time optimum five fractional-order values from all fractional-order values. Accordingly, the results for the five values q= [0.01, 0.05, 0.07, 0.09 and 0.1] are given in Fig. 9, respectively. When Fig. 9 is examined, when q = 0.01, the system synchronizes in 5.2 (time unit) time, while at q = 0.1, the system synchronizes in 1.7 (time unit). Fig. 10 shows the graph showing P-C synchronization times with different q values of the fractional-order Burke-Shaw chaotic system. The effect of the change in q value in Fig. 9 on the synchronization time can be seen more clearly on the graph in Fig. 10. When Fig. 10 is examined, it clearly shows that the increase in q value leads to a decrease in synchronization time. Accordingly, the optimum fractional-order value was determined as 0.1 because the system synchronized with a 1.7 (time unit) value in the shortest time.  Table 1 shows the data showing the behavior of the fractional-order Burke-shaw chaotic attractor given in (12) at different fractional-order values. According to Table 1, the fractional-order Burke-shaw chaotic attractor shows its chaotic features at the intervals of 0.01≤q≤0.1 and 0.8≤q≤0.99, while it emerges from its chaotic feature in the range of 0.2≤q≤0.7. Accordingly, the fractional-order value is in the range of 0.2≤q≤0.7, the system emerges from chaotic behavior and this situation affects P-C synchronization and does not provide synchronization. When the already sensitive structure of the Burke-Shaw chaotic system works with fractal values at high values, it causes the system to exit chaotic behavior.  Table 2, the synchronization times of the Burke-Shaw chaotic attractor and active control, P-C method, and fractional-order P-C method methods are compared. Accordingly, the active control method obtains 4 (time units) as the synchronization time. Integer-order P-C method obtained a value of 6 (time unit) during the synchronization period. When fractional-order is used, it is synchronized at 5.2 (time unit) when q = 0.01, 3 (time unit) when q = 0.05, 2.1 (time unit) when q = 0.07, 1.85 (time unit) when q = 0.09, and at 1.7 (time unit) when q=0.1. According to the results, the P-C method, which uses the optimum fractional-order value, achieves very successful results compared to the active control method.