Hidden Dynamics Investigation, Fast Adaptive Synchronization, and Chaos-Based Secure Communication Scheme of a New 3D Fractional-Order Chaotic System

Abstract

In this paper, a new fractional-order chaotic system containing several nonlinearity terms is introduced. This new system can excite hidden chaotic attractors or self-excited chaotic attractors depending on the chosen system parameters or its fraction-order derivative value. Several dynamics of this new system, such as chaotic attractors, equilibrium points, Lyapunov exponents, and bifurcation diagrams, are analyzed analytically and numerically. Then, adaptive control laws are developed to achieve chaos synchronization in two identical new systems with uncertain parameters; one of these two new identical systems is the master, and the other is the slave. In addition, update laws for estimating the uncertain slave parameters are derived. Furthermore, in chaos application fields, these master and slave synchronized systems are applied in secure communication to act as the transmitter and receiver, respectively. Finally, the security analysis metric tests were analyzed using histograms and spectrograms to establish the communication system’s security strength. Numerical test results demonstrate the possibility of using this proposed fractional-order chaotic system in high-security communication systems. The employed communication system is also highly resistant to pirate attacks.

1. Introduction

In 1892, Henri Poincaré introduced the first understanding of the chaos possibility [1]. Nonlinear phenomena in chaos have been used extensively in engineering, science, and applied mathematics over the past few decades [2]. Chaos occurs when a deterministic system exhibits the strange phenomenon of aperiodic trajectories. When the initial conditions and system parameters have a strong influence on chaotic systems, chaotic system sensitivity is introduced; however, even minor changes in the initial conditions can have a significant impact on the final result [3]. As result, the chaotic system is considered a complex system.

Recent research has concentrated on chaotic systems with hidden attractors. Since the seminal paper by Leonov et al., attractors in dynamical systems have been classified as self-excited attractors and hidden attractors [4]. A basin of attraction is associated with an unstable equilibrium of a self-excited attractor. On the other hand, an attractor is said to be hidden if its basin of attraction does not intersect with any of the unstable equilibrium’s small neighborhoods [5]. Chaotic attractors in dynamical systems with no equilibrium, surfaces or lines of equilibria, and stable equilibria are also classified as latent attractors [6].

Chaotic complex systems with hidden attractors are very important in many application fields of science and engineering, such as induction motors for drilling [7], bridge wing design [8], chemical reactor systems [9], aircraft control systems [10], encryption [11], PLL systems [12], weather prediction systems [13], memristive circuits [14], and secure communication schemes [15].

The fractional-order derivative and fractional-order integration calculus have recently gained popularity because fractional calculus offers more accurate models than integer-order calculus [16]. The fractional-order chaotic models exhibit more complex dynamical behavior than their integer analogs and thus play an important role in the fields of secure communication applications and high-level cryptography [17]. Moreover, because the fractional-order derivative value (q) of the system can be changed in the (0, 1] range, that gives an additional system parameter as well as the original system parameters [18].

Through the development and applications of chaos theory, many fractional-order chaotic systems have been introduced, such as the fractional-order Lorenz system [19], fractional-order Rössler system [20], fractional-order Chen system [21], fractional-order Liu chaotic system [22], fractional-order financial systems [23], fractional-order Lotka–Volterra system [24], and various other systems [25]. In the past few decades, the control and synchronization of chaotic systems have attracted much attention, and chaos synchronization presents the major issue of applying chaotic systems to secure communication systems and encryption systems [26]. Various control and synchronization techniques for chaotic systems have been developed in recent years such as sliding mode control [27], adaptive control [28], the active control method based on parameter identification [29], passive control [30], impulsive control [31], and so on as in [32]. In the secure communication application field, synchronization is achieved between two chaotic systems; the first one is a master, and the second response one is a slave, where the master presents the transmitter side, and the slave acts as the receiver side [33]. The researchers employed many fractional-order chaotic systems, and they were used in secure communication and encryption [34].

In this work, a new three-dimensional fractional-order chaotic system is proposed. The proposed system displays more complex dynamic properties than an analog integer chaotic system, which can be considered the main reason for using these systems in the design of high-security communication and encryption systems [35]. In addition, the proposed system does not contain complex functions such as the hyperbolic tangent function. Thus, it can be implemented simply using an electronic circuit. The system dynamics, such as system equilibria, chaotic attractors, bifurcation diagrams, and Lyapunov exponents, are studied analytically and numerically. Because the new system is being used in a secure communication system, an adaptive synchronization mechanism is established between two identical new systems, one of which is acting as the master (transmitter) and the other as the slave (receiver). In the adaptive synchronization process, the master adaptive control laws that will drive the slave were derived based on Lyapunov theory. Consequently, the update laws that are responsible for estimating the uncertain slave parameter correspond to the equivalent master parameters. Our work, tests, and results were verified by using the MATLAB platform. Several chaotic-based safe communication schemes have been implemented in recent years. The main approaches for utilizing chaotic signals in secure communications are chaotic masking, inclusion, chaotic shift keying, and parameter modulation. The chaotic-masking approach has the benefit of being simple and straightforward to use in electronic circuits [36]. Therefore, this technique has been employed in our work.

The rest of this paper’s organization is as follows: The fundamentals of fractional-order calculus are introduced in Section 1. In Section 2, the new fractional-order chaotic system is presented, and its chaotic attracter’s classes are determined; in addition, the system equilibria and corresponding eigenvalues are investigated in this section. The system dynamics, including Lyapunov exponents, and a bifurcation diagram are considered in Section 3. In Section 4, we establish the synchronization strategy between two identical new systems, where the adaptive control laws and the update laws are derived based on Lyapunov stability. In Section 5, the achieved synchronization is applied to the construction of the secure communication scheme, where the master and slave perform the role of transmitter and receiver, respectively. In Section 7, histograms and spectrograms are examined to demonstrate the security analysis of the communication strategy in use. Finally, the conclusions are presented in Section 8.

2. Fractional Calculus Fundamentals

Fractional calculus is a mathematical extension of integer-order calculus [37]. While it incorporates the benefits of integer-order calculus, it has its own logic and laws as well. The Riemann–Liouville (RL), Caputo, and Grünwald–Letnikov (GL) concepts are generally used in the definitions of fractional calculus [38]. In 1832, Liouville introduced the first study on FOC, and in 1867, Grünwald presented his work on the fractional-order system. In the year 1892, Riemann completed the theoretical analysis and development of the fractional-order system [39]. The basic function used in fractional-order calculus known as the Gamma function is noted by the following, which is defined as in [40]:

$$\Gamma \left(k\right)={\int }_0^{+\infty }{\mathrm{e}}^{-t}{t}^{k-1}\mathrm{d}t;k>0$$

(1)

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

The fractional integral operator (J^q) introduced by Riemann–Liouville for order (q ≥ 0) of function (f(t)) is given by [41].

$$J^{q}f\left(t\right)=\left\{\begin{array}{c}\frac{1}{\Gamma \left(q\right)}{\int }_0^t{\left(t-s\right)}^{q-1}f\left(t\right)s ; q<0\\ f\left(t\right) ; q=0\end{array}\right..$$

(2)

The fractional-order calculus form by Caputo is expressed as follows (3) [42]:

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

(3)

where m is the first integer number more than q-order.

Grünwald–Letnikov approximates the fractional derivative as described below in Equation (4) [43]:

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

(4)

where h is the step size.

3. New 3D Fractional-Order Chaotic System

Fractional-order chaotic systems are a special form of nonlinear chaotic systems; in addition, this form of a system has the basic characteristics of an integer-order chaotic system, and it also has other characteristics such as extra complexity and comfortable behavior [44]. Despite the fact that many novel chaotic systems have been presented in recent years, developing, discovering, and analyzing new chaotic systems is still beneficial to the subject of chaos in theoretical and practical fields. This fact is stated in numerous specialized papers, some of which are cited in this article’s sources [45,46,47,48]. The proposed new three-dimensional fractional-order chaotic system with two quadratic nonlinearity terms is described by the fractional-order dynamics in Equation (5).

$$\frac{{\mathrm{d}}^{q}x}{\mathrm{d}{t}^q}=z+xy; \frac{{\mathrm{d}}^{q}y}{\mathrm{d}{t}^q}=a-bz-x^2; \frac{{\mathrm{d}}^{q}z}{\mathrm{d}{t}^q}=-x+cy$$

(5)

In system (5), the system state variables are symbolized by x, y, and z, while a, b, c, and d are symbolized for the system parameters, whereas the q symbol represents the fraction-order derivative value. The system parameters have the main effects on the system behaviors, where these parameters can be constant and/or integer parameters [49,50,51]. The equilibria (equilibrium points) of this system can be obtained by equalizing the system Equations to zero as in the following Equation.

$$z+xy=0; a-bz-x^2=0;-x+cy=0$$

(6)

If the parameter a of this system is considered to be 1 (a = 1), then the equilibrium points of the system (5) are computed to be as below.

$$x=\pm \sqrt{\frac{-a}{\left(\left(b/c\right)-1\right)}} ;y=\frac{1}{c}x;z=-\frac{1}{c} x^2$$

(7)

It is clear that there are two cases for the solutions of Equation (6) that are determined by the selected parameter values of the parameters b and c.

3.1. Case1: Hidden Attractors

Regarding the equilibrium points obtained by Equation (7), if (b/c) ≥ 1, that yields two sets of solutions as in the following: If b > c that results in x² = $\sqrt{-Ve}$, so x = ±i (imaginary value), so there are no equilibrium points. On the other hand, if b = c or/and c = 0, that leads to a contradiction in the solution of Equation (7); in other words, there are no solutions for Equation (7), so, there are no equilibrium points in the system (5) in this situation also. As a result of that, the suggested system (5) can excite hidden chaotic attractors in this case.

The proposed system parameters are chosen as a = 1, b = 0.05, and c = 0.05, and the fractional-order derivative value (q = 0.98) with the initial condition of (x₀, y₀, z₀) = (0.1, 0.1, 0.1) and the corresponding hidden attractor projections are shown in Figure 1. The simulation results were obtained for a time of 500 s with a step size of h = 0.005 using the Roberto Garrappa method of solving fractional-order nonlinear systems [52].

3.2. Case2: Self-Excited Attractor

When (b/c) < 1, the fractional chaotic system (3) has two equilibrium points.

$$E_{1,2}=(\pm \sqrt{\frac{-a}{\left(\left(b/a\right)-1\right)}} , \pm \frac{1}{c}\sqrt{\frac{-a}{\left(\left(b/a\right)-1\right)}},-\frac{1}{c} \left(\frac{-a}{\left(\left(b/a\right)-1\right)} \right) .$$

The corresponding Jacobian matrix of the fixed points is obtained as follows.

$$J_{E_{1,2}}=\left[\begin{array}{ccc} \pm \frac{1}{c}\sqrt{\frac{-a}{\left(\left(b/a\right)-1\right)}},& \pm \sqrt{\frac{-a}{\left(\left(b/a\right)-1\right)}} & 1\\ -2\left(\pm \sqrt{\frac{-a}{\left(\left(b/a\right)-1\right)}}\right)& 0& –b\\ –1& c& 0\end{array}\right]$$

(8)

The eigenvalues for the Jacobian matrix can be calculated using the following Equation (9) to assess the stability of the equilibrium points.

$$\left|\lambda I-J_E\right|$$

(9)

If our proposed system parameters are selected as a = 1, b = 0.05, and c = 0.1, that causes the equilibrium points to be E_1,2 = (±√2, ±10√2, −20) and the corresponding eigenvalues as λ_1,2,3 = (13.7781, 0.3903, −0.0263). As a result, the equilibrium points are unstable foci. When the used fractional-order derivative value is (q = 0.98) and the initial conditions are (x₀, y₀, z₀) = (0.1, 0.1, 0.1), the corresponding self-excited attractor projections are shown in Figure 2.

4. Dynamical Behavior of New System

In this section, the system is analyzed based on Lyapunov exponents and bifurcation diagrams, where MATLAB is used to numerically explore the bifurcation diagrams and the Lyapunov exponents.

4.1. Lyapunov Exponents

In order to have an improved observation of the hidden chaotic attractor, the Lyapunov exponents were determined. The Lyapunov exponents were calculated with respect to time and to fractional-order derivative value (q) as shown in Figure 3. It can be seen from Figure 3a that the new system excites a hidden chaotic attractor without any equilibrium point because there is a positive Lyapunov exponent. The system parameters are selected as a = 1, b = 0.05, and c = 0.05, and the fractional-order derivative value is (q = 0.98) with the initial conditions as (x₀, y₀, z₀) = (0.1, 0.1, 0.1), and then the corresponding obtained Lyapunov exponents are Le1 = 0.2643, Le2 = −0.1223, and Le3 = −0.2372. In Figure 3b, the same above-mentioned parameters of Figure 3a are used, except that the fractional-order derivative value is changing as q ∈ [0.8, 1]. The corresponding calculated Lyapunov exponents are Le1 = 0.0203, Le2 = −0.0049, and Le3 = −0.0007, which indicates that the system can excite a hidden attractor without any equilibria because a positive Lyapunov exponent is verified [53]. The Lyapunov exponents are explored according to [54].

4.2. Bifurcation Diagrams

Finally, the bifurcation diagrams were numerically determined to further demonstrate that the new system exhibits chaotic behavior. The bifurcation diagrams were determined with respect to the fractional-order derivative value (q) and the system parameters (a, b, and c), where these diagrams were obtained by plotting the local maximum of state variable z(t) with the varying fractional-order derivative value (q) and the system parameters (a, b, and c). Figure 4 presents the bifurcation diagram with the selected initial conditions as (x₀, y₀, z₀) = (0.1, 0.1, 0.1). From Figure 4a, it can be noted that the system excites a hidden chaotic attractor as the fractional-order derivative value changes as q ∈ [0.8, 1]. Further, other classes of chaotic attracter are excited when q = 0.98 and the system parameters were changed as in Figure 4b–d.

5. Synchronization Strategy

The synchronization of two new fractional-order chaotic systems that are identical to one another is investigated in this section. So that the slave (response system) and master system (drive system) trajectories asymptotically match and confirm synchronization, a synchronization mechanism is explored. Chaos synchronization is commonly used in secure communication applications. Due to chaos’ unique properties, such as time-based complexity, extreme initial conditions sensitivity, and unpredictable chaos behavior, it is a particularly secure approach to hiding information in a chaotic signal, resulting in a transmission of a secure encrypted signal. On the other hand, by employing the chaotic synchronization technique at the receiver end, the original information signal can be retrieved.

With the growth of research, many control and synchronization mechanisms have been proposed in the chaos field and especially their applications in secure communication schemes [55]. In this work, an adaptive synchronization technique is developed to synchronize two identical new systems; one acts as the drive system (the master or transmitter side), and the other acts as the response system (the slave or receiver side). The design of the synchronization controller is developed based on Lyapunov theory, then the uncertain parameters on the slave side are estimated with derived updating laws to reach the master corresponding parameters. This is responsible for achieving synchronization and reducing synchronization error as much as possible to be zero in a very short time. In addition, the response system and drive system can have different sets of initial conditions. As a result, the two identical new systems can reach a perfect synchronization between their state variables and sustain it.

5.1. Fast Adaptive Controller Design

In this subsection, a controller for the synchronization of two identical new systems based on the Lyapunov method is developed, considering the consistent slave–master systems stated in Equations (10) and (11), respectively.

$$\frac{{\mathrm{d}}^q{x}_m}{\mathrm{d}{t}^q}=z_m+x_m{y}_m; \frac{{\mathrm{d}}^q{y}_m}{\mathrm{d}{t}^q}=1-b_m{z}_m-{x_m}^2; \frac{{\mathrm{d}}^q{z}_m}{\mathrm{d}{t}^q}=-x_m+c_m{y}_m$$

(10)

$$\frac{{\mathrm{d}}^q{x}_s}{\mathrm{d}{t}^q}=z_s+x_s{y}_s+u_1; \frac{{\mathrm{d}}^q{y}_s}{\mathrm{d}{t}^q}=1-b_s\left(t\right)z_s-{x_s}^2+u_2; \frac{{\mathrm{d}}^q{z}_s}{\mathrm{d}{t}^q}=-x_s+c_s\left(t\right)y_s+u_3$$

(11)

In Equation (11), u₁, u₂, and u₃ present the adaptive synchronization controllers that need to be designed, and b_s(t) and c_s(t) are the uncertain slave system parameters that must be estimated. The master–slave synchronization errors are defined as:

$$e_x=x_s-x_m; e_y=y_s-y_m; e_z=z_s-z_m$$

(12)

Therefore, the dynamic errors are determined as in Equations (13)–(15)

$$\frac{{\mathrm{d}}^q{e}_x}{\mathrm{d}{t}^q}=e_z+y_s{z}_s-y_m{z}_m+u_1$$

(13)

$$\frac{{\mathrm{d}}^q{e}_y}{\mathrm{d}{t}^q}=-b_s\left(t\right)e_z-e_b{z}_m-{x_s}^2+{x_m}^2+u_2$$

(14)

$$\frac{{\mathrm{d}}^q{e}_z}{\mathrm{d}{t}^q}=-e_x+c_s\left(t\right)e_y+e_c{y}_m+u_3$$

(15)

where e_b and e_c are the master–slave parameter estimation errors as in Equations (16) and (17), respectively.

$$e_b=b_s\left(t\right)-b_m$$

(16)

$$e_c=c_s\left(t\right)-c_m$$

(17)

Therefore, the dynamics of the parameter estimation error can be calculated by:

$${\dot{e}}_b={\dot{b}}_s\left(t\right); {\dot{e}}_c={\dot{c}}_s\left(t\right)$$

(18)

To determine stable update laws for estimating the uncertain slave parameter, the Lyapunov method is used [56]. Selecting the quadratic positive definite Lyapunov function as:

$$V\left(e_x,e_y,e_z,e_b,e_c\right)=\frac{1}{2}\left(e_x^2+e_y^2+e_z^2+e_b^2+e_c^2\right)$$

(19)

it follows that:

$$\dot{V}=\left(e_x\frac{{\mathrm{d}}^q{e}_x}{\mathrm{d}{t}^q}+e_y\frac{{\mathrm{d}}^q{e}_y}{\mathrm{d}{t}^q}+e_z\frac{{\mathrm{d}}^q{e}_z}{\mathrm{d}{t}^q}+e_b{\dot{e}}_b+e_c{\dot{e}}_c\right)$$

(20)

Then, substituting Equations (13)–(18) in Equation (20) yields:

$$\dot{V}=e_x\left(e_z+y_s{z}_s-y_m{z}_m+u_1\right)+e_y\left(-b_s\left(t\right)e_z-e_b{z}_m-{x_s}^2+{x_m}^2+u_2\right)+e_z\left(-e_x+c_s\left(t\right)e_y+e_c{y}_m+u_3\right)+(b_s\left(t\right)-b_m){\dot{b}}_s\left(t\right)+(c_s\left(t\right)-c_m){\dot{c}}_s\left(t\right)$$

(21)

Now, the synchronization controller functions have been selected as:

$$u_1=-k_x{e}_x-e_z-y_s{z}_s+y_m{z}_m ; u_2=-k_y{e}_y+b_s\left(t\right)e_z+{x_s}^2-{x_m}^2 ; u_3=-k_z{e}_z+e_x-c_s\left(t\right)e_y$$

(22)

where, k_x, k_y, and k_z are positive constants. In Equation (22), the uncertain slave parameters are estimated by the following update laws:

$${\dot{b}}_s\left(t\right)=z_m{e}_y; {\dot{c}}_s\left(t\right)=-y_m{e}_z$$

(23)

Finally, by substituting Equations (22) and (23) in Equation (21), we obtain:

$$\dot{V}=-k_x{e_x}^2-k_y{e_y}^2-k_z{e_z}^2$$

(24)

As noted in Equation (24), it is a negative definite function based on Lyapunov theory. Thus, based on Lyapunov stability [57], this ensures that the synchronization state errors and the master–slave parameter estimation error converge to zero exponentially with respect to time for any initial conditions.

5.2. Numerical Simulation Results

In this section, computer simulations are achieved to prove the efficacy of the introduced synchronization technique. These numerical simulations are performed using the MATLAB platform. In the simulation, the parameters of the systems in Equations (10) and (11) are chosen as b_m = 0.4, c_m = 0.4, b_s(t) is uncertain, c_s(t) is uncertain, and the fractional-order derivative value q = 0.9 with the initial conditions for the master and slave being (x_m(0), y_m(0), z_m(0)) = (0.1, 0.1, 0.1) and (x_s(0), y_s(0), z_s(0)) = (−1, 1, −0.5), respectively.

The obtained synchronized states are shown in Figure 5. Figure 6 confirms that the synchronization errors e_x, e_y, and e_z converge to zero quickly with time (in less than 0.3 s). In addition, the accurate estimation of the uncertain slave parameters with the update laws in Equation (23) is demonstrated in Figure 7. From Figure 7, it is clear that the uncertain slave parameters b_s(t) and c_s(t) are estimated correctly corresponding to the master parameters b_m = 0.4 and c_m = 0.4, respectively. The estimation process is verified rapidly with time.

6. Chaos-Based Secure Communication System

Fractional-order chaotic systems may provide more efficient secure communication when compared with traditional chaotic systems; this is because the derivative order(s) can be observed as additional system parameter(s) as well as the basic system parameter(s). Synchronization in chaotic systems has potential applications in the secure communication fields and control processing. The aforementioned introduced adaptive synchronization technique is used in secure communication arrangements. Many chaotic systems have been developed and used in a secure communication system based on different synchronization techniques as in [58,59,60,61,62]. Based on the developed adaptive synchronization technique between the two identical new fractional-order systems (10) and (11), the secure communication scheme is demonstrated by the block diagram as shown in Figure 8. In Figure 8, the main two parts of the used secure communication system are the transmitter (which presents the master or drive system) and the receiver (which presents the slave or response system).

Many techniques, including chaotic shift keying, chaotic masking, inclusion, and parameter modulation, have been used to implement secure communication based on chaos. The most popular technique is the chaotic-masking technique, and this technique has been employed in our work.

The chaotic-masking method applied to the transmitter adds the information signal to the chaotic generator state output to produce an encrypted signal that is transmitted to the receiver [63]. To hide the information signal and achieve synchronization at the receiver, the information signal needs to be 20 to 30 dB less than the chaotic generator output signal. As a result, in order to recover the original information signal at the receiver, the matching regenerated chaotic signal is subtracted. At the transmitter side, the information signal m(t) is added to the chaotic output state signal (y_m), as in the following Equation:

$$s\left(t\right)=m\left(t\right)+y_m\left(t\right)$$

(25)

where s(t) denotes the acquired masked signal that will be sent to the receiver. Thus, by subtracting the output chaotic state signal (y_s) from the encrypted signal as in Equation (26), the original information signal can be recovered. The information signal in the numerical simulation is given by Equation (27) as below.

$$m\left(t\right)=Asin\left(wt\right)$$

(26)

In this Equation (26), the information signal is sinusoidal with amplitude A and frequency w (rad/sec). The obtained numerical simulation results of our chaos-based secure communication system are shown in Figure 9, where the parameters are selected as a_m = a_s = 1, b_m = 4, and c_m = 0.4, and b_s and c_s are estimated to correspond to the matched bm and cm, respectively, while q = 0.98 and the initial conditions are (0.1, 0.1, 0.1) and (1, 0, −0.5) for the transmitter and receiver, respectively. According to Equation (27), one can retrieve the original information signal m(t) as below.

$$\widehat{m}\left(t\right)=s\left(t\right)-y_s\left(t\right)$$

(27)

7. Communication System’s Security Analysis

The secure communication system is well-renowned for its resistance to pirate attacks. To illustrate the high-security performance of the applied communication system in resisting pirate attacks, histograms and spectrograms were used in this study, because the presence of noise makes identifying a signal more challenging. As a result, the applicable communication system’s security was studied, where the original signal with A = 0.05 and w = 100 rad/s was tested.

7.1. Histogram Analysis

A signal’s histogram is a graph that shows how signal intensity levels are distributed. The statistical analysis of the original data and the encrypted versions can both be evaluated using histogram analysis [64]. Because the original information signal is a sine-wave signal, it has significant intensities on the sine-wave peaks. The histogram of an encrypted signal, on the other hand, shows a wide range of intensity levels. An encrypted signal’s histogram should differ from the original signal’s histogram both statistically and visually. The original information signal, encrypted signal, and recovered signal histograms are depicted in Figure 10.

When comparing the histograms in Figure 10a,b, it is obvious that the encrypted signal histograms have a significantly different distribution than the original signal, proving the applicable communication system’s great robustness against statistical attack. Furthermore, the encrypted image’s histogram distribution in Figure 10c is identical to that of the original image in Figure 10a. As a result, the original image can be retrieved effectively.

7.2. Spectrogram Analysis

A spectrogram is yet another effective technique for calculating the spectrum of time signals that is employed in a variety of applications. A spectrogram is a plot that depicts the frequency of a signal on the vertical axis, time on the horizontal axis, and signal power on a color scale to offer information about power as a function of frequency and time [65]. The spectrograms of the original, encrypted, and recovered signals are shown in Figure 11. The spectrogram was computed using the short-time Fourier transform (STFT).

8. Conclusions

A new fractional-order chaotic system is presented in this work. The behavior of the dynamics of this proposed system containing the chaotic attractors, the fixed points and corresponding eigenvalues, the bifurcation diagrams, and the Lyapunov exponents have been considered. As it is clear, our suggested fractional-order chaotic system is simple to implement, where the suggested system did not contain complex functions (for example, hyperbolic tangent functions). In our system, all mathematical functions are addition, subtraction, and multiplication; these functions can easily be implemented with simple analog op-amps like summer, subtracter, and multiplier, respectively. The investigated dynamics prove that the new FOCS system can excite hidden chaotic attractors or self-excited chaotic attractors. On the basis of the results of the simulation, it can be concluded that the dynamical behavior of the proposed system is very complex and extremely sensitive to the system parameters and initial conditions and to slight variations in the value of the fractional-order derivative. In addition, the proposed system is highly suited for usage in high-security communication systems because it offers nonlinear dynamical characteristics of incredible complexity.

Following this, an adaptive synchronization approach was developed. This synchronization strategy was established between two identical new systems, one of which serves as the master and the other as the slave. The adaptive control laws which are responsible for verifying the synchronization were derived on the basis of the Lyapunov stability approach. Correspondingly, the uncertain slave update parameter laws were determined to estimate these uncertain parameters. Additionally, in the chaos application field, a secure communication scheme was constructed on the basis of the developed synchronization technique. Finally, the security analysis metric tests were examined with histograms and spectrograms to determine the security strength of the employed communication system. The obtained experimental results and thorough security evaluations validate the efficiency, high security, and time efficiency of the employed cryptosystem and demonstrate a high, robust resistance to attacks.