A New Fractional-Order Chaotic System with Its Analysis, Synchronization, and Circuit Realization for Secure Communication Applications

Abstract

This article presents a novel four-dimensional autonomous fractional-order chaotic system (FOCS) with multi-nonlinearity terms. Several dynamics, such as the chaotic attractors, equilibrium points, fractal dimension, Lyapunov exponent, and bifurcation diagrams of this new FOCS, are studied analytically and numerically. Adaptive control laws are derived based on Lyapunov theory to achieve chaos synchronization between two identical new FOCSs with an uncertain parameter. For these two identical FOCSs, one represents the master and the other is the slave. The uncertain parameter in the slave side was estimated corresponding to the equivalent master parameter. Next, this FOCS and its synchronization were realized by a feasible electronic circuit and tested using Multisim software. In addition, a microcontroller (Arduino Due) was used to implement the suggested system and the developed synchronization technique to demonstrate its digital applicability in real-world applications. Furthermore, based on the developed synchronization mechanism, a secure communication scheme was constructed. Finally, the security analysis metric tests were investigated through histograms and spectrograms analysis to confirm the security strength of the employed communication system. Numerical simulations demonstrate the validity and possibility of using this new FOCS in high-level security communication systems. Furthermore, the secure communication system is highly resistant to pirate attacks. A good agreement between simulation and experimental results is obtained, showing that the new FOCS can be used in real-world applications.

1. Introduction

In the last few decades, the nonlinear phenomenon in chaos has been widely considered in engineering, sciences, and applied mathematics [1,2]. When a deterministic system exhibits a strange phenomenon of aperiodic trajectories, this is called chaos [3,4,5]. Chaotic system sensitivity is introduced when the initial conditions and system parameters have a strong influence on chaotic systems; however, even a minor alteration in the starting situation can have a significant impact on the final result [6].

Chaos plays a significant role in many fields, such as biological systems dealing with the human heart and brain [7], protected communication systems [8], smart systems [9], digital signal processing [10], data encryption [11], robotics engineering [12], adaptive control engineering [13], and nonlinear oscillator designs [14]. Recently, several fractional-order systems are proposed to have exhibit chaotic behavior, such as a fractional-order Lotka-Volterra system, a fractional-order Chua Hartley oscillator, a fractional-order Van der Pol oscillator [15], a fractional-order Lui system [16], a fractional-order Lorenz system [17], and many other systems [18]. The FOCSs are very useful in secure communication and high-level encryption because they exhibit very high complexity in their dynamics.

Fractionality is a branch of mathematics that is an extension of classical calculus. The study of fractional calculus has recently received a lot of attention due to its potential application in different fields [19]. In 1694, Leibniz was asked by L’Hospital if the half order derivative is possible [20]. Representation of a system by a fractional order gives several benefits over its classical integer-order depiction. The fractional-order has transmissible properties, and as a result of that, the core of the real physical system can be more exactly maintained. In addition, the system order “q” can be varied in a range (0, 1] that allows an extension in the system parameter space [21,22,23].

Fractional calculus has applications in a wide range of engineering and science fields, including fluid mechanics, viscoelasticity, systems’ identification, electromagnetics, electrochemistry, biological population models, signal processing, optics, control, analog filters, circuit theory, oscillators, encryption systems, image processing, economics, and chemistry [24,25,26]. Because fractional-order chaotic models include the fractional-order parameter as well as the original system features, they have more complex dynamical behavior than integer models [27,28].

The chaos synchronization technique is based on the notion that two chaotic systems might evolve on different attractors, but when they are synchronized, they start on various attractors and eventually follow the same course. The synchronization between two systems can be achieved when the trajectories of two systems are matched [29,30,31]. In chaos systems, the synchronization mechanism plays a significant role in constructing secure communication schemes [32,33,34]. It can occur between two chaotic systems; one is a master and the driven one is a slave [35,36]. In such applications, the master grants the transmitter, and the slave demonstrates the receiver. Chaotic synchronization was first studied by Yamada and Fujisaka in 1983 [37]. Many control methods have been developed for controlling and synchronizing the fractional-order chaotic systems, such as active control, impulsive control, adaptive control, passive control, and sliding mode control [38].

Many FOCSs have been recognized and used in the security of communication systems. Jialin Hou et al. [39] used the FOCS for proposing image encryption. They use exclusive or (XOR) for encryption algorithm; this encryption system has increased the arbitrariness and improved the encryption speed. The authors developed a traditional integer Chen chaotic system to be a fractional type. Therefore, this system cannot be considered a novel in the subject area. As a result, the encryption system did not provide more high-security performance. Maitreyee Dutta et al. [40] suggested a novel FOCS. The suggested system shows multiple wing chaotic attractors which produce chaos for generating a change in complex attractors by changing only one parameter. A synchronization mechanism based on an adaptive sliding mode controller is used in that system. The proposed system contains a hyperbolic tangent function, which increases the difficulty of its implementation in the real world. On the other hand, our proposed system has not contained such these functions. Thus, it can be implemented simply. Zahra Rashidnejad et al. [41] introduced a finite time synchronization mechanism of two different FOCSs master and slave systems with unknown parameters, disturbances, and uncertainties, where the synchronization was rapidly realized in perfect time. The realization of this investigated synchronization mechanism was not taken into account to prove the feasibility of the application in the real world. An improved cryptosystem image based on FOCS has been proposed by Musheer Ahmad et al. with better performance and appropriateness of the proposed improved algorithm for realizing a robust and strong safe communication scheme [42]. In this work, a control method by Pecora and Carroll (PC) was used to achieve the synchronization mechanism. In our work, we used the adaptive synchronization mechanism. The adaptive synchronization technique has many advantages, including rapid dynamics responses, good transient performance, and a robust technique for parameter variations, initial conditions, and disturbances.

Several chaotic-based safe communication schemes have been applied over the last few years. Chaotic masking, inclusion, chaotic shift keying, and parameter modulation are the main approaches for exploiting chaotic signals in secure communications [43,44,45].

In this work, the chaotic masking technique was employed, and a new four-dimensional FOCS was proposed. The proposed FOCS exhibits more complex attractors than an integer regular chaotic system complexity which makes it very suitable for designing high-security communication systems. Since the new FOCS was applied in the secure communication scheme, an adaptive synchronization mechanism was developed between two identical FOCSs, one presented as the master (transmitter) and the other for the slave (receiver). Adaptive control laws were derived based on the Lyapunov approach, which are responsible for attaining master–slave synchronization. Furthermore, an update control rule was developed to estimate an uncertain parameter in the slave side according to the equivalent master side parameter. For verifying the security analysis of the used communication system, histograms and spectrograms were explored. The achieved results were tested and determined by using MATLAB Simulink (MathWorks, Portola Valley, California, United States). Furthermore, a feasible electronic circuit of that new FOCS and its synchronization was realized by Multisim.

The paper’s organization is as follows. In Section 2, the fundamental calculus of fractional order is described. In Section 3, the new FOCS are studied by equilibria, eigenvalues, and their chaotic attractors. Thy dynamical analysis including the fractal dimension and Lyapunov exponents bifurcation diagrams are investigated in Section 4. For attaining chaos synchronization between two new identical FOCSs with an uncertain parameter in the slave side, adaptive control laws are derived in Section 5. In Section 6, we realize an electronic circuit of the new FOCS. In Section 7, the proposed FOCS and the developed synchronization mechanism are digitally implemented, where the Arduino Due board is used. In Section 8, the suggested FOCS and the synchronization process are applied for encoding and decoding a stream of binary numbers in the master and the slave sides, respectively. In Section 9, histograms and spectrograms are investigated for proving the security analysis of the utilized communication scheme. In Section 10, we conclude this paper.

2. Fractional Order Preliminaries

The history of fractional calculus extends back more than three hundred years [46]. In the development of fractional calculus, there are several fundamentals for differentiation and integration. The popularly used definitions of fractional-order calculations were developed by Grünwald–Letnikov, Caputo, and Riemann–Liouville [47].

Generally, in fractional-order chaotic systems modeling, the Caputo derivative is preferred due to several advantages. Firstly, it takes into account the influences of initial conditions. Secondly, it has a clearer physical meaning. Finally, all fractional operators take the memory effect into account [48,49]. Caputo introduced a definition of the fractional-order derivative (q-order) for a function f(t), as expressed in the following Equation (1) [50].

$$t_0{D}_t^{q}f\left(t\right)=\left\{\begin{array}{c}\frac{1}{\Gamma \left(m-q\right)}\underset{t_0}{\overset{t}{{\displaystyle \int }}}\frac{f^{\left(m\right)}\left(\tau \right)}{{\left(t-\tau \right)}^{q-m+1}}d\tau ; m-1<q<n\\ \frac{d^{m}f\left(t\right)}{d{t}^m} ; q=m.\end{array}\right.$$

(1)

where m is the least integer number, greater than q and $\Gamma \left(m-q\right)$ is known as Gamma function is the function that is most used in the fractional calculus, and it is defined in Equation as follows (2) [51]:

$$\Gamma \left(x\right)=\underset{0}{\overset{+\infty }{{\displaystyle \int }}}{e}^{-t}{t}^{x-1}dt ; x>0$$

(2)

With $\Gamma \left(1\right)=1;\Gamma \left(0\right)=+\infty$.

Grunwald–Letnikov approaches the fractional derivative, as demonstrated in the following [52].

$$D^{q}x\left(t\right)=f\left(x,t\right)=\underset{h\to 0}{\lim }{h}^{-q}\left(-1\right){\displaystyle \sum }_{j=0}^{t/h}\left(-1\right)\left(\begin{array}{c}q\\ j\end{array}\right)x\left(t-jh\right)$$

(3)

where h is the step size.

The fractional integral operator (J^q) created by Riemann–Liouville for order (q ≥ 0) of function (f(t)) is given by Equation (4) [53].

$$J^{q}f\left(t\right)=\left\{\begin{array}{c}\frac{1}{\Gamma \left(q\right)}\underset{0}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{q-1}f\left(t\right)d\tau ; q<0,\\ f\left(t\right); q=0.\end{array}\right.$$

(4)

3. New Fractional-Order Chaotic System

There are two types of fractional order systems: the commensurate fractional-order system and the incommensurate fractional-order system. The fractional-order values (q₁, q₂,...q_n) of the system equations are equaled in the first type (q₁ = q₂ =...=q_n), while these values are unequaled in the second type (q₁ ≠ q₂ ≠, …≠q_n) [54].

The FOCS are a special type of nonlinear system; this type of system has the original properties of integer-order chaotic systems and has additional properties, such as high complexity and extreme sudden behavior. Despite the fact that there are a lot of new chaotic systems proposed over the last few years, it is still advantageous for the field of chaos in theoretical and practical areas to develop, discover, and analyze new chaotic systems. This fact is stated in many specialized articles and the article’s references [55,56,57,58] are some of these examples. That is because some of the chaotic applications, such as secure communication, require new systems continuously. Here, we proposed a new 4-D FOCS, as demonstrated by the fractional-order dynamic Equation (5).

$$\begin{gathered} \frac{d^{q}x}{d{t}^q}=-ayz, \\ \frac{d^{q}y}{d{t}^q}=bxz-c, \\ \frac{{\mathrm{d}}^{q}z}{d{t}^q}=x-dz, \\ \frac{d^{q}w}{d{t}^q}=k{x}^3-w. \end{gathered}$$

(5)

In Equation (5), x, y, z, and w are the state variables; a, b, d, and k are system positive constant parameters; and q is the fractional-order derivative value. The dynamics which are chaotic attractors, fixed (equilibrium) points, and Lyapunov exponent are investigated for the proposed system. The system (5) shows chaotic actions when selecting parameters a = 2.5, b = 0.05, c = 1.2, d = 2, k = 0.001, and the fractional-order q = 0.9. Where the initial conditions are x(0) = 0.1, y(0) = 0.2, z(0) = −1, and w(0) = 0.3. The phase portraits of the proposed system are illustrated in Figure 1. The simulation results for systems (5) were obtained using the Roberto Garrappa method of solving fractional order nonlinear systems, with a step size of (h = 0.005) [59].

The equilibrium points (equilibria) of the novel proposed fractional-order chaotic system (5) can be obtained by solving the following nonlinear Equations.

$$\frac{{\mathrm{d}}^{q}x}{\mathrm{d}{t}^q}=-ayz=0;\frac{{\mathrm{d}}^{q}y}{\mathrm{d}{t}^q}=bxz-c=0;\frac{{\mathrm{d}}^{q}z}{\mathrm{d}{t}^q}=x-dz=0;\frac{{\mathrm{d}}^{q}w}{\mathrm{d}{t}^q}=k{x}^3-w=0$$

(6)

The equilibrium points are determined as given in

Table 1. Equilibrium points, consistent eigenvalues, and equilibria class.

No.	Equilibrium Point	Eigenvalues	Equilibria Classification
1	(6.9282, 0, 3.4641, 0.3326)	(−1, 0.2052 + 1.5643i, 0.2052 − 1.5643i, −2.4104)	Saddle focus
2	(−6.9282, 0, −3.4641, −0.3326)	(−1, 0.2052 + 1.5643i, 0.2052 − 1.5643i, −2.4104)	Saddle focus

. In addition, the eigenvalues consistent with the obtained equilibrium points are calculated by linearizing method (Jacobian matrix of system (5)), as described in Equation (7).

$$J=\left[\begin{array}{cc}\begin{array}{cc}0& -az\\ bz& 0\end{array}& \begin{array}{cc}-ay& 0\\ bx& 0\end{array}\\ \begin{array}{cc}1 & 0\\ 3k{x}^2& 0\end{array}& \begin{array}{cc}-d& 0\\ 0& -1\end{array}\end{array}\right]$$

(7)

The equilibrium point in fractional-order systems is stable if the following criterion is met:

$$\left|arg\left({\lambda }_i\right)\right|>q\frac{\pi }{2} ; i=1,2,3$$

(8)

Thus, the equilibria of the new FOCS were classified according to the condition in (8), as illustrated in Table 1 [60]. Generally, a dynamic system with two saddle equilibria connected by heteroclinic exhibits chaos [61]. That is exactly verified in the proposed new nonlinear fractional-order systems.

4. The Dynamical Analysis

Generally, the major dynamical tools that may be used to examine the dynamical behaviors of nonlinear chaotic systems are the fractal dimension, bifurcation diagrams, and Lyapunov exponents [62,63,64]. The fractal dimension, bifurcation diagrams, and Lyapunov exponents are numerically investigated using MATLAB.

4.1. Fractal Dimension

The basic parameter for describing self-similar patterns and processes is the fractal dimension. The concept of fractal dimension is useful in describing natural objects because it indicates their degree of complexity [65]. There are several approaches to calculate the fractal dimension. For example, Higuchi’s method, the rescaled range method, Renyi’s entropy, and Katz’s method. Higuchi’s fractal dimension (HFD) is a popular nonlinear measure for signal analysis and it is a time-domain signal complexity metric [66].

Higuchi’s method for calculating the fractal dimension of a signal or function is explained following [67]. An epoch with N number of samples of the signal is represented by x(1), x(2), ..., x(N). K is new sub-epochs that are constructed from this original epoch and represented by $X_m^k$, where sub-epoch is described by the following equation.

$$\begin{gathered} X_m^k=\left\{x\left(m\right),\right.x\left(m+k\right),x\left(m+2k\right),\dots ,x\left(m+int\left[\left(N-k\right)/k\right]k\right);m=1,2,\dots ,k \\ k=1,2,\dots k_{max} \end{gathered}$$

(9)

In Equation (9) m, k, int(.), and k_max are the initial time, the time interval, integer real part number, and a free parameter, respectively.

The average length $L_m\left(k\right)$ for each of the sub-epochs constructed can be calculated: as:

$$L_m\left(k\right)=\frac{1}{k}\left[\left({\displaystyle \sum }_{i=1}^{int\left[\left(N-m\right)/k\right]}\left|x\left(m+ik\right)-x(m+\left(i-1\right)k\right|\right)\frac{N-1}{\left(int\left[\left(N-m\right)/k\right]\right)k}\right]$$

(10)

The mean of the k values is used to calculate the length of the epoch L(k) for the time interval k as follows.

$$L\left(k\right) =\frac{1}{k}{\displaystyle \sum }_{m=1}^k{L}_m\left(k\right)$$

(11)

Finally, Higuchi’s fractal dimension can be estimated as.

$$HFD=\frac{ln(L\left(k\right) }{ ln\left(1/k\right)}$$

(12)

The choice of the free parameter k_max is critical in HFD estimation. In our work, HFD values were calculated for various k_max values with N ranging from 500 to 10,000, as shown in Figure 2. A smooth signal’s HFD value (for example, a low-frequency sine wave and linear) was estimated to be 1. The HFD of random white noise or more complex signals was calculated to be ≈2. Because it is a numerical approximation of the fractal dimension, the calculated HFD values may be slightly greater than 2 [68]. Therefore, it can be noted from Figure 2 that the suggested FOCS exhibits very complex dynamical behavior.

4.2. Bifurcation Diagrams

It can be proven that the proposed system exhibits the chaoticity phenomena by the bifurcation diagrams [69]. For the proposed system, the bifurcation diagram is numerically calculated in two cases, one with respect to parameter d of the system in (5), and the other case versus the fractional-order derivative value (q).

Figure 3a shows the system bifurcation diagram against the parameter d, where the parameters are selected as a = 2.5, b = 0.05, c = 1.2, k = 0.001, and the fractional order q = 0.9 with initial conditions x(0) = 0.1, y(0) = 0.2, z(0) = −1, and w(0) = 0.3. Figure 3b shows the system bifurcation diagram against the fractional order (q) with the same values in Figure 3a, except the parameter d = 2. The new system exhibits chaos in range d ∈ [0.745, 2] and, for the fractional-order derivative value (q), the range is [0.896, 1]. As seen in Figure 3, the new proposed system exhibits a variety of bifurcation topological patterns.

4.3. Lyapunov Exponents

Lyapunov exponents are calculated and strongly indicate that the new system exhibits the chaoticity phenomena. At least one positive Lyapunov exponent in nonlinear dynamic systems ensures that these systems display chaos [70,71]. For the suggested system, the Lyapunov exponents are numerically determined, as shown in Figure 4. The Lyapunov exponents are obtained as: Le1 = 0.1585, Le2 = −0.0001, Le3 = −1.0366, and Le4 = −2.2034. Thus, the proposed FOCS exhibits chaotic performance because one of the Lyapunov exponents is positive.

Moreover, Lyapunov exponents are determined when the fractional-order derivative value is changed to q ∈ [0.88, 1], as illustrated in Figure 5. The system parameters utilized in this calculation are the same as those used in the first Lyapunov exponent’s computations. As shown in Figure 5, the Lyapunov exponents are Le1 = 0.6461, Le2 = 0.1654, Le3 = −0.1223, and Le4 = −0.7161, which indicates a chaotic phenomenon in our proposed system.

5. Adaptive Synchronization

Practically, the two important ways to utilize chaos are chaos control and synchronization techniques [72]. The chaos synchronization has received much attention; this is because the secure communication schemes primarily depend on the synchronization between the transceivers [73,74]. Researchers have introduced many chaos control and synchronization techniques, as listed in the introduction section.

In this work, an adaptive synchronization mechanism is developed for synchronizing two identical new FOCSs; one acts the master (transmitter side) and the other acts the slave (receiver side). Based on the Lyapunov method, adaptive control laws were derived, which are responsible for achieving the synchronization between the master and the slave. Furthermore, an update control law was designed for estimating the single uncertain parameter in the slave side, corresponding to the equivalent master parameter.

5.1. Adaptive Control Law Design

In this section, we derive the adaptive and update control laws. The drive system (master) and controlled system (slave) dynamics are named as in Equations (13) and (14), respectively:

$$\frac{d^q{x}_m}{d{t}^q}=-a{y}_m{z}_m; \frac{d^q{y}_m}{d{t}^q}=b{x}_m{z}_m-c; \frac{d^q{z}_m}{d{t}^q}=x_m-d_1{z}_m; \frac{d^q{w}_m}{d{t}^q}=k{x}_m{}^3-w_m$$

(13)

$$\frac{d^q{x}_s}{d{t}^q}=-a{y}_s{z}_s+u_1+u_1; \frac{d^q{y}_s}{d{t}^q}=b{x}_s{z}_s-c+u_2; \frac{d^q{z}_s}{d{t}^q}=x_s-d_2\left(t\right)z_s+u_3; \frac{d^q{w}_s}{d{t}^q}=k{x}_s{}^3-w_s+u_4$$

(14)

As shown in Equation (14), the adaptive controllers to be designed are u₁, u₂, u_3, and u_4, and d₂(t) is the uncertain parameter in the slave system to be estimated. The synchronization errors between systems (13) and (14) can be defined in the following equation.

$$e_1=x_s-x_m; e_2=y_s-y_m; e_3=z_s-z_m; e_4=w_s-w_m$$

(15)

Therefore, the error dynamics can be obtained by Equations (16)–(19).

$$\frac{d^q{e}_1}{d{t}^q}=a\left(y_m{z}_m-y_s{z}_s\right)+u_1$$

(16)

$$\frac{d^q{e}_2}{d{t}^q}=b\left(x_s{z}_s-x_m{z}_m\right)+u_2$$

(17)

$$\frac{d^q{e}_3}{d{t}^q}=e_1-d_2{e}_3-e_d{z}_m+u_3$$

(18)

$$\frac{d^q{e}_4}{d{t}^q}=k(x_s{}^3-x_m{}^3)-e_4+u_4$$

(19)

In Equation (18), the master–slave parameter estimation error e_d(t) is given in Equation (20).

$$e_d= d_2\left(t\right)-d_1$$

(20)

As a result, the parameter error dynamics can be obtained in Equation (21).

$${\dot{e}}_d={\dot{d}}_2\left(t\right)$$

(21)

The Lyapunov approach is used to derive the update law that is responsible for estimating the uncertain parameter in the slave side corresponding to the equivalent parameter of the master. The quadratic Lyapunov function is determined in Equation (22).

$$V\left(e_1,e_2,e_3,e_4,e_d\right)=\frac{1}{2}\left(e_1^2+e_2^2+e_3^2+e_4^2+e_d^2\right)$$

(22)

Then, differentiating Equation (22) with respect to time, yields Equation (23).

$$\dot{V} =\left(e_1\frac{{\mathrm{d}}^q{e}_1}{\mathrm{d}{t}^q}+e_2\frac{{\mathrm{d}}^q{e}_2}{\mathrm{d}{t}^q}+e_3\frac{{\mathrm{d}}^q{e}_3}{\mathrm{d}{t}^q}+e_4\frac{{\mathrm{d}}^q{e}_4}{\mathrm{d}{t}^q}+e_d{\dot{e}}_d\right)$$

(23)

Replacing the dynamic errors from Equations (16)–(21) in Equation (23) gives:

$$\begin{gathered} \dot{V} =\left(e_1\left(a\left(y_m{z}_m-y_s{z}_s\right)+u_1\right)+e_2\left(b\left(x_m{z}_m-x_s{z}_s\right)+u_2\right)+e_3\left(e_1 \\ -d_2{e}_3-e_d\left(t\right)z_m+u_3\right)+e_4\left(k(x_s{}^3-x_m{}^3\right)-e_4+u_4\right) \\ +\left(d_2\left(t\right)-d_1\right){\dot{d}}_2\left(t\right) \end{gathered}$$

(24)

The feedback adaptive controller functions have been designed, as named in Equation (25) based on Lyapunov theory for stability. Where many methods can be used for designing similar controllers, as in [75,76].

$$\begin{gathered} u_1=-k_1{e}_1-\left(1+a\left(y_m{z}_m-y_s{z}_s\right)e_1 ; u_2 \\ =-k_2{e}_2-\left(1+b\left(x_m{z}_m-x_s{z}_s\right)\right)e_2 ; u_3 \\ =-k_3{e}_3-e_1+d_2{e}_3 -e_3; u_4 \\ =-k_4{e}_4-k(x_s{}^3-x_m{}^3\right)e_4 \end{gathered}$$

(25)

where k₁, k₂, k₃ and k₄, are positive constants. Additionally, the updating law that is responsible for updating the uncertain slave parameter is obtained in Equation (26):

$${\dot{e}}_d=z_m{e}_3$$

(26)

Finally, by substituting the feedback adaptive controller functions described by Equation (25) and the update law named by Equation (26) in Equation (24), we get Equation (27).

$$\dot{V} =-k_1{e}_1{}^2-k_2{e}_2{}^2-k_3{e}_3{}^2-k_4{e}_4{}^2$$

(27)

A negative definite function is shown in Equation (27) [77]. Therefore, for any initial conditions, the synchronization state errors converge to zero exponentially with respect to time. In addition, the estimated slave parameter exponentially aligned with the equivalent master parameter. As a result, the established synchronization technique’s asymptotic global stability is guaranteed.

5.2. Numerical Simulation

For the numerical simulations, MATLAB software is used for simulating this adaptive synchronization mechanism between the two identical FOCSs (13) and (14). In the synchronization simulation, the parameters of the new FOCSs are chosen as a = 2.5, b = 0.05, c = 1.2, d₁ = 2, d₂(t) is uncertain, k = 0.001, and the fractional order q = 0.9. Where the initial conditions for the master are x_m(0) = 0.1, y_m(0) = 0.2, z_m(0) = −1, and w_m(0) = 0.3. For the slave, they are x_s(0) = 0.2, y_s(0) = 0, z_s(0) = 0.1, and w_s(0) = −1.

The state variables synchronization of the identical novel FOCSs is shown in Figure 6. The 3-D projections of phase portrait for the master–slave synchronization are demonstrated in Figure 7, where the synchronization process was simulated for 12 s only. Figure 8 confirms the convergence of the synchronization errors e₁, e₂, e₃, and e₄ which exponentially tend to zero with respect to time. Figure 9 illustrates the parameter estimation, where parameter d₂(t) in the slave side is exponentially converging to the equivalent parameter value (d₁ = 2) on the master side.

Furthermore, the slave was subjected to disturbances for testing the robustness and disturbance rejection capabilities of the developed synchronization mechanism. Where the disturbances δ₁(t), δ₂(t), δ₃(t), and δ₄(t) were inserted into the slave state variables, as shown in the following equation.

$$\begin{gathered} \frac{d^q{x}_s}{d{t}^q}=-a{y}_s{z}_s+u_1+{\delta }_1\left(t\right); \frac{d^q{y}_s}{d{t}^q}=b{x}_s{z}_s-c+u_2-a{y}_s{z}_s+u_1+{\delta }_2\left(t\right); \\ \frac{d^q{z}_s}{d{t}^q}=x_s-d_2\left(t\right)z_s+u_3-a{y}_s{z}_s+u_1+{\delta }_3\left(t\right); \\ \frac{d^q{z}_s}{d{t}^q}=x_s-d_2\left(t\right)z_s+u_3-a{y}_s{z}_s+u_1+{\delta }_4\left(t\right) \end{gathered}$$

(28)

The developed synchronization mechanism shows a good robustness against disturbances as confirmed in Figure 10. The disturbances are chosen as δ₁(t) = 0.15sin(4t), δ₂(t) = 0.08sin(4t), δ₃(t) = 0.1sin(4t), and δ₄(t) = 0.13sin(4t). Where these disturbances were subjected to slave states at a time of 25 s.

6. Electronic Circuit Realization

In real-world applications, there are no electronic circuit devices that can reproduce the fractional-order operator; therefore, there are challenges in realizing the fractional-order transfer functions. Ahmed and Sprott introduced the approximation of 1/s^0.9, as in Equation (29) [78].

$$\frac{1}{s^{0\cdot 9}}= \frac{2.2675\left(s+1.292\right)\left(s+215.4\right)}{\left(s+0.01292\right)\left(s+2.154\right)\left(s+359.4\right)}$$

(29)

The electrical component that exhibits fractional order impedance features is called a fractance [79]. In realization of these devices, the classical integer integrator (op-amp integrator) can be used by replacing the capacitor with the fractance that is equivalent to the designed fractional-order operator. Therefore, the fractional-order integrator is obtained. The realization of the proposed FOCS and its synchronization by an electronic circuit are verified for confirming the feasibility of using the new FOCS in real applications.

In the circuit design of the FOCS, for the order q = 0.9, the key idea is how to design the circuit of 1/s^0.9. The chain fractance realization corresponding to 1/s^0.9 is shown in Figure 11 [80].

So, based on the fractional-order frequency domain calculation function in Equation (29), it is easy to construct a FOCS electronic circuit. It needs simple elements, such as operational amplifiers and resistors, as well as the chain fractance element that is shown in Figure 11 above. Figure 12 illustrates the electronic circuit schematic diagram for realizing the proposed FOCS. In Figure 13, the phase portraits of the chaotic attractors for the realized electronic circuit of the new FOCS are obtained. Additionally, the adaptive synchronization mechanism of two FOCSs is verified by electronic circuits. The results of electronic circuits of the master–slave system synchronization are shown in Figure 14. Circuit simulation outputs (see Figure 13) show good qualitative matching with the numerical simulations (see Figure 1).

7. Microcontroller Implementation

The main purpose of this section is to prove the implementation of the new FOCS and the developed synchronization mechanism in real-world applications. Fractional-order chaotic systems can be implemented in hardware using several embedded devices, such as Raspberry Pi, microcontrollers, DSP boards, FPGA boards, and analog electronics circuit implementation, as described in Section 6.

In our work, we used a microcontroller (Arduino Due) to implement the new FOCS system in Equation (5) using a discrete technique, as described in [81]. The Arduino Due is a digital board that features an Atmel SAM3X8E processor and an ARM Cortex-M3 processor. It has a structure that is suited for executing complex arithmetic operations. It features the following characteristics in a nutshell: 4 UARTs, 2 DAC (digital to analog), power jack 2 TWI, SPI header, JTAG header, 32-bit ARM core microcontroller, 84 MHz clocks, 54 digital input/output pins, 12 analog inputs, USB OTG capable connection, 2 DAC (digital to analog), and power jack 2 TWI.

The following are the main benefits of using this microcontroller: 12-bit resolution for its two peripherals, DAC0 and DAC1, and a lower cost when compared with FPGAs or DSP boards. The Arduino IDE uses an Arduino-specific programming language, which is comparable with C++ to program the ARM microcontroller via the native USB port (serial port) [82]. Figure 15 shows the Arduino Due microcontroller setup for implementing the new FOCS.

For the experimental test, the FOCS parameters are selected as a = 2.5, b = 0.05, c = 1.2, d = 2, k = 0.001, and the fractional-order q = 0.9 with the following initial conditions: x(0) = 0.1, y(0) = 0.2, z(0) = −1, and w(0) = 0.3. The ADC0 and DAC1 are utilized to generate phase portraits of chaotic attractors by analog oscilloscope for the system described by Equation (5), as shown in Figure 16.

Because the digital to analog converters (DAC0 and DAC1) on the microcontroller operate between 0.5 and 2.7 V, the amplitudes of the MATLAB simulation and the Arduino Due experimental results for the state variables (system trajectories) of the proposed FOCS will be different. To reach the same amplitude values as the estimated numerical simulations, an external operational amplification step would be required.

Furthermore, the developed synchronization technique was digitally implemented and tested by the Arduino Due board. The main advantages satisfied when the synchronization process is implemented by that microcontroller type include its open source, s the simplicity of the implementation, and the lower cost when compared with other above-mentioned devices.

Figure 17 shows the behavior of the synchronization error obtained experimentally in the oscilloscope when compared to the synchronization errors obtained by numerical simulation in Figure 8. It is clear that they have the same behavior, implying that the developed adaptive master–slave synchronization process can be digitally implemented by Arduino Due boards. Where the FOCS parameters, initial conditions, and the fractional-order derivative value are used as selected in the numerical simulation, as described by Section 5.2.

Figure 18 shows the experimental results of parameter estimation, where the slave side’s parameter d₂(t) is exponentially convergent to the master side’s equivalent parameter value (d₁ = 2).

8. Application in Secure Communication

Due to the broadband spectrum of chaotic signals, it is very suitable to use these systems in secure communication applications [83]. The above designed adaptive synchronization technique is applied in secure communication arrangements. The chaotic masking technique has the advantage of being simple and easy to implement by electronic circuits [84]. Therefore, this technique was used in our work. Figure 19 depicts the block diagram of the suggested secure communication scheme based on an adaptive synchronization technique between two new FOCSs. As mentioned above, the overall construction is mainly composed of two parts: the derived end (master or transmitter) and controlled end (slave or receiver).

As shown in Figure 19, the information signal m(t) is added to the generated chaotic signal from the transmitter side, which results in the transmitted signal c(t) being encoded. Thanks to the synchronization, at the receiver side, the same chaotic signal is created at the transmitter side. Then, the original information signal can be recovered by subtracting the chaotic signal from the encoded signal.

On the transmitter side, the information signal is added to the generated chaotic signal as in Equation (30).

$$c\left(t\right)= m\left(t\right)+w_m\left(t\right)$$

(30)

On the other hand, at the receiver, the original information signal $\widehat{m}\left(t\right)$ can be retrieved by subtracting the receiver chaotic signal from the transmitted signal as in Equation (31). Figure 20 illustrates the simulation results of a secure communication scheme based on FOCS, where a noiseless channel is assumed.

$$\widehat{m}\left(t\right)= c\left(t\right)-w_s\left(t\right)$$

(31)

In the real world, a certain amount of noise is introduced to the transmitted signals in the real channel. Therefore, additive white Gaussian noise (AWGN) with various signal-to-noise ratio (SNR) values is considered in this work. The white Gaussian noise symbolled by n(t) displays in the transmitted signal as follows.

$$c\left(t\right)= m\left(t\right)+w_m\left(t\right)+n\left(t\right)$$

(32)

The obtained results of the recovered signal with SNR up to 30 dB and 35 dB are shown in Figure 21a,b respectively. As seen from Figure 21, the distortion noticed within the recovered signal is further increased in the noise level (decrease in the SNR).

9. Security Analysis

It is well known that a secure communication system should be resistant to pirate attacks. Histograms and spectrograms were used in this work to demonstrate the high-security performance of the applied communication system in resisting pirate attacks. This is because the presence of noise increases the difficulty of detecting a signal. Therefore, the security analysis of the applied communication system was investigated, with the assumption of a noiseless channel.

9.1. Histogram Analysis

The histogram of any signal is a graph that depicts the distribution of signal intensity levels [85]. Histogram analysis can be used to evaluate the statistical analysis of the original information and the encrypted versions [86]. Because an information signal is a stream of binary numbers, it only has two clear intensities. As a result, the histogram will display two levels indicating the signal distribution among the intensity values. On the other hand, the histogram of encrypted signal displays many different levels of its intensities.

The histogram of an encrypted signal should be statistically and visually different from the histogram of the original signal. Figure 22 depicts the histograms of the original information signal, an encrypted signal, and a recovered signal.

When comparing the histograms in Figure 22a,b, it is clear that the distribution of the encrypted signal histograms differs significantly from that of the original signal, demonstrating the applied communication system’s high robustness against statistical or attacks. In addition, the histogram distribution for the encrypted image in Figure 22c is similar to that of the original image in Figure 22a. Thus, the original image can be successfully recovered.

9.2. Spectrogram Analysis

A spectrogram is another powerful technique for estimating the spectrum of time signals that is used in many applications [87]. A spectrogram is a plot that shows the frequency of the signal on the vertical axis, time on the horizontal axis, and signal power on a color scale to provide information about power as a function of frequency and time [88,89]. Figure 23 shows spectrograms of the original, encrypted, and recovered signals. Where the short-time Fourier transform (STFT) is used to compute the spectrogram.

Figure 23 demonstrates that the original information signals differ significantly from the encrypted signals. The recovered signal, on the other hand, is similar to the original signal. As a result of these findings, high resistance against pirate attacks is provided by the employed communication system. In addition, the original image can be successfully restored.

10. Conclusions

A new FOCS is presented in this work. The dynamics of the FOCS, including the chaotic attractors, the fractal dimension, bifurcation diagrams, and Lyapunov exponent, were investigated, proving that the system exhibits very complex dynamic behaviors. Then, an adaptive synchronization technique was proposed. This technique was verified between two identical FOCSs (master and slave). Moreover, and in order to confirm the feasibility of using this new system in real applications, an analog electronic chaotic circuit was realized and examined for the proposed FOCS and its synchronization. Additionally, a microcontroller (Arduino Due) was used to demonstrate the feasibility of implementing the suggested system and the developed synchronization mechanism in real-world applications. Furthermore, the developed synchronization mechanism was used for constructing a secure communication system. Finally, the security analysis metric tests were analyzed using histograms and spectrograms to establish the communication system’s security strength. The simulation results revealed that the suggested FOCS system’s dynamic behavior is extremely sensitive to initial conditions, system parameters, and fractional-order derivative values (q). The simulation results obtained from MATLAB are consistent with the experimental results (analog and digital), indicating that the system is capable of being employed in real-world applications. The main advantages of using Arduino Due board to implement this system and the developed synchronization mechanism are the simplicity and low cost of implementation compared to other devices, such as FPGAs and DSPs, which are more expensive. According to the obtained results of this investigation, the proposed FOCS is simple to be implemented. Additionally, the new FOCS is very suitable to be used in high-security communication systems as it provides nonlinear dynamical behaviors with extreme complexity. The secure communication scheme is also extremely resistant to pirate attacks.