Chaos Controllability in Fractional-Order Systems via Active Dual Combination–Combination Hybrid Synchronization Strategy

Abstract

In this paper, the dual combination–combination hybrid synchronization (DCCHS) scheme has been investigated in fractional-order chaotic systems with a distinct dimension applying a scaling matrix. The formulations for the active control have been analyzed numerically using Lyapunov’s stability analysis in order to achieve the proposed DCCHS among the considered systems. With the evolution of time, the error system then converges to zero by applying a suitably designed control function. The proposed synchronization technique depicts a higher degree of complexity in error systems, and therefore, the DCCHS scheme provides higher protection for secure communication. Mathematical simulations are implemented using MATLAB, the results of which confirm that the proposed approach is superior and more effective in comparison to existing chaos literature.

1. Introduction

Specifically, fractional calculus is a more generalized version of calculus, unlike the integer-order calculus for the study of non-linear dynamical systems. This concept of fractional differentiation was first raised by French mathematician Guillaume de L’Hôpital in the 17th century on 30 September 1695. Fractional-order differential systems have many applications in various branches, such as biological models [1], image processing [2], robotics [3], information processing [4], and finance models [5], etc.

Prominently, chaos synchronization and control have an immensely wide spectrum of applications in applied science, engineering, and technology, such as in secure communication [6], neural networks [7], image encryption [8], ecological models [9], and so on. Thus far, numerous types of secure communication schemes have been introduced [10,11,12,13], such as chaos modulation [11,14,15,16], chaos shift keying [17,18], and chaos masking [13,16,19]. In chaos communication strategies, the simple key idea of transmitting a message via chaotic/hyperchaotic models is that a message signal is embedded in the transmitter model that initiates a chaotic/disturbed signal. Then, this disturbed signal is emitted to the receiver through a universal channel. The message signal is finally recovered by the receiver. Chaotic models have been essentially utilized both as transmitters and receivers. Subsequently, this field of chaos synchronization and control has sought significant deliberation among varied research fields.

By now, several techniques to attain the chaos synchronization phenomenon have been reported and discussed, such as complete synchronization [20], anti-synchronization [21], hybrid synchronization [22], combination projective synchronization [13], dual combination synchronization (DCS) [23], double compound synchronization [24], combination–difference synchronization [1], and hybrid projective combination–combination synchronization (CCS) [25]. Dual synchronization is a specialized characteristic of chaos synchronization wherein four identical and non-identical chaotic systems (two masters along with two slave systems) are synchronized. In combination synchronization, two master systems are synchronized with one slave system. The CCS is an extension of combination synchronization, which is written as a combination of two chaotic systems as master systems and a combination of two chaotic systems as slave systems that are synchronized. Although the dual combination–combination synchronization (DCCS) of fractional-order chaotic models has huge potential and significance in contrast to CCS, these vital observations have prompted the authors to analyze the DCCHS in fractional-order (FO) chaotic systems.

For chaos control and synchronization, there are different methods that have been utilized in existing chaos literature, such as active control [26], adaptive control [27], adaptive sliding mode control (ASMC) [6], etc. Chaotic systems are essentially nonlinear dynamic systems that are extremely sensitive to initial conditions. In addition, a hyperchaotic (HC) system is basically a chaotic behavior having at least 2 +ve Lyapunov exponents; the smallest number of dimensions for HC is four. Historically, the first 4D HC was introduced in 1979 by Rössler [28]. Furthermore, generalizing the 3D Lorenz model to a 5D Lorenz model by changing the state variable to complex one was first introduced by Fowler et al. [29] in 1982. The synchronization of the combination of two master systems and two slave systems are discussed in [30]. In addition, the authors [31] discussed the phase CCS among FO chaotic systems. The DCS for FO in a complex system has been presented in [23]. In addition, the authors [32] discussed the dual multi-switching CCS technique. The authors [13] studied combination projective synchronization in fractional-order chaotic systems with disturbance and uncertainty. In addition, the authors [33] proposed dual synchronization of FO chaotic systems using a linear controller. Furthermore, the authors [34] studied parameter identification and the finite-time C–C synchronization. Moreover, a DCS technique with different dimensions was announced by the authors [35]. The authors [36] used active control to perform a dual combination–combination anti-synchronization scheme.

Based on the aforesaid observations, we investigate the DCCHS scheme between non-identical, FO, complex chaotic systems and FO hyperchaotic systems with different dimensions using scaling matrices. In the DCCHS technique, four master systems, namely, the Lorenz system, T-system, Lu system, and Chen system, have been considered. In addition, four corresponding slave systems are the hyperchaotic Xing, Vanderpol, Rabinovich, and Rikitake systems. The responding controllers are outlined in view of Lyapunov’s stability analysis (LSA) and the efficient active control method.

The main features of our proposed work in the current manuscript are:

• The proposed DCCHS methodology considers eight non-identical, complex, FO chaotic/ hyperchaotic systems.
• It describes a robust DCCHS scheme-based controller to achieve dual combination–combination hybrid synchronization in considered systems and conducts oscillation in synchronization errors with fast convergence.
• The designing of the active controllers is carried out in a simplified manner using LSA and a master–salve configuration.
• Simulation outcomes alongside a table showing a comparison analysis demonstrate the efficacy of the introduced methodology.

The remainder of the paper is described as follows: Section 2 contains few essential preliminary results which will be used in the following sections. Section 3 deals with the problem formulation of the DCCHS scheme. Section 4 consists of an example illustrating the DCCHS scheme via the active control methodology. Section 5 exhibits numerical simulation with a discussion to establish the effectiveness and suitability of the considered DCCHS scheme. Lastly, Section 6 provides few significant concluding remarks.

2. Preliminaries

Definition 1 

([37]). The Caputo’s fractional derivative of order $q\in R^{+}$ for function $\Psi \left(t\right)$ is defined by:

$$\begin{array}{c}\hfill c{D}_t^q\Psi \left(y\right)=\frac{1}{\gamma (n-q)}{\int }_c^t\frac{{\Psi }^n\left(y\right)}{{(t-y)}^{q-n+1}}dy,\end{array}$$

(1)

where $n-1<q<n$ and $\gamma \left(q\right)={\int }_0^{\infty }{\psi }^{q-1}{e}^{-\psi }d\psi$ is the gamma function.

Property 1. 

If $\Psi \left(t\right)$ is a constant function and of the order $q>0$, the Caputo fractional-order derivative satisfies the conditions:

$$D^q\Psi \left(t\right)=0.$$

Property 2.

The Caputo derivative satisfies the following linear property:

$$D^q[c_1{\Psi }_1\left(t\right)+c_2{\Psi }_2\left(t\right)]=c_1{D}^q{\Psi }_1\left(t\right)+c_2{D}^q{\Psi }_2\left(t\right),$$

where ${\Psi }_1\left(t\right)$ and ${\Psi }_2\left(t\right)$ are functions of t, and $c_1$ and $c_2$ are constants.

Definition 2 

([35]). We introduce the stability results of the linear FO differential system. Given the autonomous system:

$$\begin{array}{c}\hfill D_t^{q}w\left(t\right)=Pw\left(t\right),\end{array}$$

(2)

where the state variable $w={(w_{11},w_{12},\cdots ,w_{1m})}^T$ and initial value $w\left(0\right)=w_0=(w_{10},w_{20},\cdots ,w_{m0})$, with $q\in (0,1)$ and $P\in R^{m\times m}$, then the system (2) is said to be asymptotically stable if and only if $|arg\left({\lambda }_i\left(P\right)\right)|>{\displaystyle \frac{q\pi }{2}}$, $i=1,2,\cdots ,m$, where $arg\left({\lambda }_i\left(P\right)\right)$ stands for the argument of the eigenvalues ${\lambda }_i$ of P.

3. Problem Formulation

The authors [23] introduce the scheme of dual combination–combination hybrid synchronization (DCCHS). Firstly, two master systems are taken as:

$$\begin{array}{cc}\hfill D^q{w}_{1m}& =p_1{w}_{1m}+f_1\left(w_{1m}\right)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill D^q{z}_{1m}& =s_1{z}_{1m}+g_1\left(z_{1m}\right).\hfill \end{array}$$

(4)

Consider the next two master systems as:

$$\begin{array}{cc}\hfill D^q{w}_{2m}& =p_2{w}_{2m}+f_2\left(w_{2m}\right)\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill D^q{z}_{2m}& =s_2{z}_{2m}+g_2\left(z_{2m}\right).\hfill \end{array}$$

(6)

For the master systems, there are two slave systems which are taken as:

$$\begin{array}{cc}\hfill D^q{w}_{3s}& =p_3{w}_{3s}+f_3\left(w_{3s}\right)+{\varphi }_1\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill D^q{z}_{3s}& =s_3{z}_{3s}+g_3\left(z_{3s}\right)+{\theta }_1.\hfill \end{array}$$

(8)

Consider the next two slave systems as:

$$\begin{array}{cc}\hfill D^q{w}_{4s}& =p_4{w}_{4s}+f_4\left(w_{4s}\right)+{\varphi }_2\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill D^q{z}_{4s}& =s_4{z}_{4s}+g_4\left(z_{4s}\right)+{\theta }_2,\hfill \end{array}$$

(10)

where $p_i\in R^{n_1\times n_1}$, $s_i\in R^{n_1\times n_1}$, ($i=1,2$) are linear parts of the systems described in (3)–(6); $w_{im}={[w_{i1m},w_{i2m},...w_{i{n}_{1}m}]}^T$, ($i=1,2$) are the state vectors of the master systems (3) and (5); $z_{im}={[z_{i1m},z_{i2m},...z_{i{n}_{1}m}]}^T$ are the state vector of the master systems (4) and (6); $p_i\in R^{n_2\times n_2}$, $s_i\in R^{n_2\times n_2}$, ($i=3,4$) are the linear parts of the systems of (7)–(10); $w_{is}={[w_{i1s},w_{i2s},...w_{i{n}_{2}s}]}^T$, ($i=3,4$) are the state vectors of the slave systems (7) and (9); $z_{is}={[z_{i1s},z_{i2s},...z_{i{n}_{2}s}]}^T$, ($i=3,4$) are the state vector of the slave systems (8) and (10); $f_i:R^{n_1}\to R^{n_1}$, ($i=1,2$) are vector functions whose values are real and are nonlinear parts of systems (3) and (5); $g_i:R^{n_1}\to R^{n_1}$, ($i=1,2$) are vector functions whose values are real and are nonlinear parts of systems (4) and (6); $f_i:R^{n_2}\to R^{n_2}$, ($i=3,4$) are vector functions whose values are real and are nonlinear parts of systems (7) and (9); $g_i:R^{n_2}\to R^{n_2}$, ($i=3,4$) are vector functions whose values are real and are nonlinear parts of systems (8) and (10); and ${\varphi }_i$ and ${\theta }_i$, ($i=1,2$) are control functions of the slave systems (7)–(10).

Define the error state functions as follows:

$$\begin{array}{cc}\hfill E_1& =(w_{3s}+w_{4s})-{\rho }_1(w_{1m}+w_{2m})\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill E_2& =(z_{3s}+z_{4s})-{\rho }_2(z_{1m}+z_{2m}).\hfill \end{array}$$

(12)

The error dynamics system turns out to be:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{D}^q{E}_1=\hfill & p_3{E}_1-p_3{w}_{4s}+p_3{\rho }_1(w_{2m}+w_{1m})+f_3\left(w_{3s}\right)+p_4{E}_2-p_4{w}_{3s}\hfill \\ & -p_4{\rho }_1(w_{1m}+w_{2m})+f_4\left(w_{4s}\right)-{\rho }_1[p_1{w}_{1m}+f_1\left(w_{1m}\right)+p_2{w}_{2m}\hfill \\ & +f_2\left(w_{2m}\right)]+{\varphi }_1+{\varphi }_2\hfill \\ D^q{E}_2=\hfill & s_3{E}_2-s_3{z}_{4s}+s_3{\rho }_2(z_{2m}+z_{1m})+g_3\left(z_{3s}\right)+s_4{E}_2-s_4{z}_{3s}\hfill \\ & +s_4{\rho }_2(z_{1m}+z_{2m})+g_4\left(z_{4s}\right)-{\rho }_2[s_1{z}_{1m}+g_1\left(z_{1m}\right)+s_2{z}_{2m}\hfill \\ & +g_2\left(z_{2m}\right)]+{\theta }_1+{\theta }_2.\hfill \end{array}\right.\end{array}$$

(13)

Theorem 1 

([23,32]). If the control functions of the slave systems are chosen in the following manner:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\varphi }_1+{\varphi }_2=\hfill & p_3{w}_{4s}-p_3{\rho }_1(w_{2m}+w_{1m})-f_3\left(w_{3s}\right)+p_4{w}_{3s}+p_4{\rho }_1(w_{1m}+w_{2m})\hfill \\ & -f_4\left(w_{4s}\right)+{\rho }_1[p_1{w}_{1m}+f_1\left(w_{1m}\right)+p_2{w}_{2m}+f_2\left(w_{2m}\right)]\hfill \\ & +(M_1+M_2)E_1\hfill \\ {\theta }_1+{\theta }_2=\hfill & s_3{z}_{4s}-s_3{\rho }_2(z_{2m}+z_{1m})-g_3\left(z_{3s}\right)+s_4{z}_{3s}-s_4{\rho }_2(z_{1m}+z_{2m})\hfill \\ & -g_4\left(z_{4s}\right)+{\rho }_2[s_1{z}_{1m}+g_1\left(z_{1m}\right)+s_2{z}_{2m}+g_2\left(z_{2m}\right)]\hfill \\ & +(M_3+M_4)E_2,\hfill \end{array}\right.\end{array}$$

(14)

where $M_1,M_2\in R^{n_2\times n_2}$ are obtained from $p_3$ and $p_4$, respectively, also called gain matrices, and $M_3,M_4\in R^{n_2\times n_2}$ are obtained from $s_3$ and $s_4$, respectively, then, the synchronization phenomenon among eight systems (3)-(10) are achieved using DCCHS if and only if $|arg\left({\lambda }_i\right)|>{\displaystyle \frac{q\pi }{2}}$ values of $p_3+M_1+p_4+M_2$ and $q_3+M_3+q_4+M_4$.

Proof. 

In Equation (13), substitute the control function values $\varphi$ and $\theta$ from Equation (14). The error dynamics system (13) has been reduced to the following form:

$$\begin{array}{c}\hfill D^q{E}_1=(p_3+M_1+p_4+M_2)E_1\\ \hfill D^q{E}_2=(s_3+M_3+s_4+M_4)E_2.\end{array}$$

(15)

Using Definition 2, the error system (15) becomes stable asymptotically using active control if and only if each eigenvalue ${\lambda }_i$ of $(p_3+M_1+p_4+M_2)$ and $(s_3+M_1+s_4+M_2)$ fulfill $|arg\left({\lambda }_i\right)|>{\displaystyle \frac{q\pi }{2}}$ for $i=1,2,...n$. Consequently, $li{m}_{t\to \infty }\parallel E\parallel =0$, and therefore, the discussed systems attained the DCCHS scheme. □

4. Illustrative Example

In this section, we present an illustrative example to explain the considered DCCHS scheme. The mathematical model of the FO complex Lorenz system [38] can be represented as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{d^q{w}_{11m}}{d{t}^q}}\hfill & =p_{11}(w_{13m}-w_{11m})\hfill \\ {\displaystyle \frac{d^q{w}_{12m}}{d{t}^q}}\hfill & =p_{11}(w_{14m}-w_{12m})\hfill \\ {\displaystyle \frac{d^q{w}_{13m}}{d{t}^q}}\hfill & =p_{12}{w}_{11m}-w_{13m}-w_{11m}{w}_{15m}\hfill \\ {\displaystyle \frac{d^q{w}_{14m}}{d{t}^q}}\hfill & =p_{12}{w}_{12m}-w_{14m}-w_{12m}{w}_{15m}\hfill \\ {\displaystyle \frac{d^q{w}_{15m}}{d{t}^q}}\hfill & =p_{11}{w}_{13m}+w_{12m}{w}_{14m}-p_{13}{w}_{15m},\hfill \end{array}\right.\end{array}$$

(16)

where $p_{11}$, $p_{12}$, and $p_{13}$ are parameters of (16) with the parameter values chosen as $p_{11}=10$, $p_{12}=180$, and $p_{13}=1$.

Now, comparing system (16) with system (3), one finds:

$$p_1=\left[\begin{array}{ccccc}-p_{11}& 0& p_{11}& 0& 0\\ 0& -p_{11}& 0& p_{11}& 0\\ p_{12}& 0& -1& 0& 0\\ 0& p_{12}& 0& -1& 0\\ 0& 0& 0& 0& -p_{13}\end{array}\right],f_1\left(w_{1m}\right)=\left[\begin{array}{c}0\\ 0\\ -w_{11m}{w}_{15m}\\ -w_{12m}{w}_{15m}\\ w_{11m}{w}_{13m}+w_{12m}{w}_{14m}\end{array}\right].$$

The mathematical model of FO complex T system [39] can be described as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{d^q{z}_{11m}}{d{t}^q}}\hfill & =q_{11}(z_{13m}-z_{11m})\hfill \\ {\displaystyle \frac{d^q{z}_{12m}}{d{t}^q}}\hfill & =q_{11}(z_{14m}-z_{12m})\hfill \\ {\displaystyle \frac{d^q{z}_{13m}}{d{t}^q}}\hfill & =(q_{12}-q_{11})z_{11}-q_{11}{z}_{11m}{z}_{15m}\hfill \\ {\displaystyle \frac{d^q{z}_{14m}}{d{t}^q}}\hfill & =(q_{12}-q_{11})z_{12m}-q_{11}{z}_{12m}{z}_{15m}\hfill \\ {\displaystyle \frac{d^q{z}_{15m}}{d{t}^q}}\hfill & =z_{11m}{z}_{13m}+z_{12m}{z}_{14m}-q_{13}{z}_{15m},\hfill \end{array}\right.\end{array}$$

(17)

where $q_{11}$, $q_{12}$, and $q_{13}$ are parameters of (17) with the parameter values chosen as $q_{11}=2.1$, $q_{12}=30$, and $q_{13}=0.6$.

On comparing system (17) with system (4), one finds:

$$q_1=\left[\begin{array}{ccccc}-q_{11}& 0& q_{11}& 0& 0\\ 0& -q_{11}& 0& q_{11}& 0\\ (q_{12}-q_{11})& 0& 0& 0& 0\\ 0& (q_{12}-q_{11})& 0& 0& 0\\ 0& 0& 0& 0& -q_{13}\end{array}\right],g_1\left(q_{1m}\right)=\left[\begin{array}{c}0\\ 0\\ -q_{11}{z}_{11m}{z}_{15m}\\ -q_{11}{z}_{12m}{z}_{15m}\\ z_{11m}{z}_{13m}+z_{12m}{z}_{14m}\end{array}\right].$$

The mathematical model of the FO complex Lu system [40] is presented as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{d^q{w}_{21m}}{d{t}^q}}\hfill & =p_{21}(w_{23m}-w_{21m})\hfill \\ {\displaystyle \frac{d^q{w}_{22m}}{d{t}^q}}\hfill & =p_{21}(w_{24m}-w_{22m})\hfill \\ {\displaystyle \frac{d^q{w}_{23m}}{d{t}^q}}\hfill & =-w_{21m}{w}_{25m}+p_{22}{w}_{23m}\hfill \\ {\displaystyle \frac{d^q{w}_{24m}}{d{t}^q}}\hfill & =-w_{22m}{w}_{25m}+p_{22}{w}_{24m}\hfill \\ {\displaystyle \frac{d^q{w}_{25m}}{d{t}^q}}\hfill & =w_{21m}{w}_{23m}+w_{22m}{w}_{24m}-p_{23}{w}_{25m},\hfill \end{array}\right.\end{array}$$

(18)

where $p_{21}$, $p_{22}$, and $p_{23}$ are parameters of (18) with the parameter values chosen as $p_{21}=40$, $p_{22}=22$, and $p_{23}=5$.

On comparing system (18) with system (5), one finds:

$$p_2=\left[\begin{array}{ccccc}-p_{21}& 0& p_{21}& 0& 0\\ 0& -p_{21}& 0& p_{21}& 0\\ 0& 0& p_{22}& 0& 0\\ 0& 0& 0& p_{22}& 0\\ 0& 0& 0& 0& -p_{23}\end{array}\right],f_2\left(w_{2m}\right)=\left[\begin{array}{c}0\\ 0\\ -w_{21m}{w}_{25m}\\ -w_{22m}{w}_{25m}\\ w_{21m}{w}_{23m}+w_{22m}{w}_{24m}\end{array}\right].$$

The mathematical model of the FO complex Chen system [41] is defined by:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{d^q{z}_{21m}}{d{t}^q}}\hfill & =q_{21}(z_{23m}-z_{21m})\hfill \\ {\displaystyle \frac{d^q{z}_{22m}}{d{t}^q}}\hfill & =q_{21}(z_{24m}-z_{22m})\hfill \\ {\displaystyle \frac{d^q{z}_{23m}}{d{t}^q}}\hfill & =(q_{23}-q_{21})z_{21m}-z_{21m}{z}_{25m}+q_{23}{z}_{23m}\hfill \\ {\displaystyle \frac{d^q{z}_{24m}}{d{t}^q}}\hfill & =(q_{23}-q_{21})z_{22m}-z_{22m}{z}_{25m}+q_{23}{z}_{24m}\hfill \\ {\displaystyle \frac{d^q{z}_{25m}}{d{t}^q}}\hfill & =z_{21m}{z}_{23m}+z_{22m}{z}_{24m}-q_{22}{z}_{25m},\hfill \end{array}\right.\end{array}$$

(19)

where $q_{21}$, $q_{22}$, and $q_{23}$ are parameters of (19) with the parameter values chosen as $q_{21}=35$, $q_{22}=3$, and $q_{23}=28$.

Now, comparing system (19) with system (6), one finds:

$$q_2=\left[\begin{array}{ccccc}-q_{21}& 0& q_{21}& 0& 0\\ 0& -q_{21}& 0& q_{21}& 0\\ (q_{23}-q_{21})& 0& q_{23}& 0& 0\\ 0& (q_{23}-q_{21})& 0& q_{23}& 0\\ 0& 0& 0& 0& -q_{22}\end{array}\right],g_2\left(z_{2m}\right)=\left[\begin{array}{c}0\\ 0\\ -z_{21m}{z}_{25m}\\ -z_{22m}{z}_{25m}\\ z_{21m}{z}_{23m}+z_{22m}{z}_{24m}\end{array}\right].$$

The four corresponding slave systems are taken as a fractional-order hyperchaotic Xling system [42] as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{d^q{w}_{31s}}{d{t}^q}}\hfill & =p_{31}(w_{32s}-w_{31s})+w_{34s}+{\varphi }_{11}\hfill \\ {\displaystyle \frac{d^q{w}_{32s}}{d{t}^q}}\hfill & =p_{32}{w}_{31s}+w_{31s}{w}_{33s}-w_{34s}+{\varphi }_{12}\hfill \\ {\displaystyle \frac{d^q{w}_{33s}}{d{t}^q}}\hfill & =-p_{33}{w}_{33s}-p_{34}{w}_{31s}{w}_{31s}+{\varphi }_{13}\hfill \\ {\displaystyle \frac{d^q{w}_{34s}}{d{t}^q}}\hfill & =p_{33}{w}_{31s}+{\varphi }_{14},\hfill \end{array}\right.\end{array}$$

(20)

where $p_{31}=10,p_{32}=40,p_{33}=2.5,p_{34}=4$, and $q=0.95$. This considered system is hyperchaotic.

Now, comparing system (20) with system (7), one finds:

$$p_3=\left[\begin{array}{cccc}-p_{31}& p_{31}& 0& 1\\ 0& p_{33}& 0& 0\\ 0& 0& -p_{32}& 0\\ 0& 0& 0& p_{34}\end{array}\right],f_3\left(w_{3s}\right)=\left[\begin{array}{c}0\\ w_{31s}{w}_{33s}\\ -p_{34}{w}_{31s}{w}_{31s}\\ 0\end{array}\right].$$

The fractional van der Pol hyperchaotic system [43] is given by:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{d^q{z}_{31s}}{d{t}^q}}\hfill & =z_{32s}+{\theta }_{11}\hfill \\ {\displaystyle \frac{d^q{z}_{32s}}{d{t}^q}}\hfill & =-(q_{31}+q_{32}{z}_{33s})z_{31s}-(q_{31}+q_{32}{z}_{33s})z_{31s}{z}_{31s}{z}_{31s}-q_{33}{z}_{32s}+q_{34}{z}_{33s}+{\theta }_{12}\hfill \\ {\displaystyle \frac{d^q{z}_{33}}{d{t}^q}}\hfill & =z_{34s}+{\theta }_{13}\hfill \\ {\displaystyle \frac{d^q{z}_{34}}{d{t}^q}}\hfill & =-q_{35}{z}_{33s}+q_{36}(1-z_{33s}{z}_{33s})z_{34s}+q_{37}{z}_{31s}+{\theta }_{14},\hfill \end{array}\right.\end{array}$$

(21)

where $q_{31}=10,q_{32}=3,q_{33}=0.4,q_{34}=70,q_{35}=1,q_{36}=5,q_{37}=0.1$, and $q=0.95$. This system is hyperchaotic.

Now, comparing system (21) with system (8), one finds:

$$q_3=\left[\begin{array}{cccc}0& 0& 0& 0\\ -q_{31}& -q_{33}& q_{34}& 0\\ 0& 0& 0& 1\\ q_{37}& 0& -q_{35}& q_{36}\end{array}\right],g_3\left(z_{3s}\right)=\left[\begin{array}{c}0\\ -z_{31s}{z}_{33s}\\ z_{31s}{z}_{32s}\\ -z_{32s}{z}_{33s}\end{array}\right].$$

The fractional-order Rabinovich hyperchaotic system [44] is described as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{d^q{w}_{41s}}{d{t}^q}}\hfill & =-p_{41}{w}_{41s}{p}_{42}{w}_{42s}+w_{42s}{w}_{23s}+{\varphi }_{21}\hfill \\ {\displaystyle \frac{d^q{w}_{42s}}{d{t}^q}}\hfill & =p_{42}{w}_{41s}-p_{43}{w}_{42s}-w_{41s}{w}_{43s}+w_{44s}+{\varphi }_{12}\hfill \\ {\displaystyle \frac{d^q{w}_{43}}{d{t}^q}}\hfill & =-p_{44}{w}_{43s}+w_{41s}{w}_{42s}+{\varphi }_{23}\hfill \\ {\displaystyle \frac{d^q{x}_{44}}{d{t}^q}}\hfill & =-p_{45}{w}_{42s}+{\varphi }_{24},\hfill \end{array}\right.\end{array}$$

(22)

where $p_{41}=34,p_{42}=6.75,p_{43}=1,p_{44}=1,p_{45}=2$, and $q=0.95$. This system is hyperchaotic.

Now, comparing system (22) with system (9), one finds:

$$p_4=\left[\begin{array}{cccc}-p_{41}& p_{42}& 0& 0\\ p_{42}& -p_{43}& 0& 1\\ -p_{44}& 0& 0& 0\\ 0& -p_{45}& 0& 0\end{array}\right],f_4\left(w_{4s}\right)=\left[\begin{array}{c}{w}_{42s}{w}_{43s}\\ -w_{41s}{w}_{43s}\\ w_{41s}{w}_{42s}\\ 0\end{array}\right].$$

The fractional-order hyperchaotic Rikitake dynamic system [45] is described as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{d^q{z}_{41s}}{d{t}^q}}\hfill & =-q_{41}{z}_{41s}+z_{42s}{z}_{43s}-q_{42}{z}_{44s}+{\theta }_{21}\hfill \\ {\displaystyle \frac{d^q{z}_{42s}}{d{t}^q}}\hfill & =-q_{41}{z}_{42s}+z_{41s}(z_{43s}-q_{43})-q_{42}{z}_{44}+{\theta }_{22}\hfill \\ {\displaystyle \frac{d^q{z}_{43s}}{d{t}^q}}\hfill & =1-z_{41s}{z}_{42s}+{\theta }_{23}\hfill \\ {\displaystyle \frac{d^q{y}_{44}}{d{t}^q}}\hfill & =q_{44}{z}_{42s}+{\theta }_{24},\hfill \end{array}\right.\end{array}$$

(23)

where $q_{41}=1,q_{42}=1.7,q_{43}=1,q_{44}=0.7$, and $q=0.95$. This system is hyperchaotic.

On comparing system (23) with system (10), one finds:

$$q_4=\left[\begin{array}{cccc}-q_{41}& 0& 0& -q_{42}\\ -q_{43}& -q_{41}& 0& -q_{42}\\ 0& 0& 0& 0\\ 0& q_{44}& 0& 0\end{array}\right],g_4\left(z_{4s}\right)=\left[\begin{array}{c}{z}_{42s}{z}_{43s}\\ z_{41s}{z}_{43s}\\ -z_{41s}{z}_{42s}\\ 0\end{array}\right],$$

where $u_{1i}$, $v_{1i}$, $u_{2i}$, and $v_{2i}$$(i=1,2,3,4,5)$ are control functions for systems (16)–(23).

The error function will be written as:

$$\begin{array}{cc}\hfill E_1& =(w_{3s}+w_{4s})-{\rho }_1(w_{2m}+w_{1m})\hfill \\ \hfill E_2& =(z_{3s}+z_{4s})-{\rho }_2(w_{2m}+w_{1m}).\hfill \end{array}$$

(24)

Matrices ${\rho }_1$ and ${\rho }_2$ are chosen as:

$${\rho }_1={\rho }_2=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& -1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& -1& -1\end{array}\right].$$

(25)

Finally, the error function can be obtained as:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{E}_{11}=(w_{31s}+w_{41s})-(w_{21m}+w_{11m})\hfill \\ E_{12}=(w_{32s}+w_{42s})+(w_{22m}+w_{12m})\hfill \\ E_{13}=(w_{33s}+w_{43s})-(w_{23m}+w_{13m})\hfill \\ E_{14}=(w_{34s}+w_{44s})+(w_{14m}+w_{15m}+w_{24m}+w_{25m})\hfill \\ E_{21}=(z_{31s}+z_{41s})-(z_{21m}+z_{11m})\hfill \\ E_{22}=(z_{32s}+z_{42s})+(z_{22m}+z_{12m})\hfill \\ E_{23}=(z_{33s}+z_{43s})-(z_{23m}+z_{13m})\hfill \\ E_{24}=(z_{34s}+z_{44s})+(z_{14m}+z_{15m}+z_{24m}+z_{25m}),\hfill \end{array}\right.\end{array}$$

(26)

where $E_1={[E_{11},E_{12},E_{13},E_{14}]}^T$ and $E_2={[E_{21},E_{22},E_{23},E_{24}]}^T$.

Using Theorem 1, the controllers would be:

$$\begin{array}{c}\hfill {\varphi }_1+{\varphi }_2=\left[\begin{array}{c}{\varphi }_{11}+{\varphi }_{21}\\ {\varphi }_{12}+{\varphi }_{22}\\ {\varphi }_{13}+{\varphi }_{23}\\ {\varphi }_{14}+{\varphi }_{24}\end{array}\right]=\left[\begin{array}{c}-p_{11}{w}_{11m}+p_{11}{w}_{13m}-p_{31}{w}_{41s}+p_{31}{w}_{42s}\\ +w_{44s}+p_{31}(w_{11m}+w_{21m})+p_{31}(w_{12m}\\ +w_{22m})-p_{31}{E}_{11}+(w_{14m}+w_{15m}+w_{24m}\\ +w_{25m})-p_{21}{w}_{21m}+p_{21}{w}_{23m}-p_{41}{w}_{31s}\\ +p_{42}{w}_{32s}-w_{42s}{w}_{43s}+p_{41}(w_{11m}+w_{21m})\\ +p_{42}(w_{12m}+w_{22m})-p_{42}{E}_{12}\\ \\ p_{11}{w}_{12m}-p_{11}{w}_{14m}+p_{32}{w}_{41s}-w_{44s}\\ -w_{31s}{w}_{33s}-p_{32}(w_{11m}+w_{21m})-(w_{14m}+w_{15m}\\ +w_{24m}+w_{25m})+p_{21}{w}_{22m}-p_{21}{w}_{24m}\\ +p_{42}{w}_{31s}-p_{43}{w}_{32s}+w_{41s}{w}_{43s}-p_{42}(w_{11m}\\ +w_{21m})-p_{43}(w_{12m}+w_{22m})+(w_{14m}\\ +w_{15m}+w_{24m}+w_{25m})-(p_{32}+p_{42})E_{11}\\ \\ p_{12}{w}_{11m}-w_{13m}-w_{11m}{w}_{15m}-p_{33}{w}_{43s}+\\ p_{34}{w}_{31s}{w}_{31s}+p_{33}(w_{13m}+w_{23m})+p_{22}{w}_{23m}\\ -w_{21m}{w}_{25m}-p_{44}{w}_{33m}-w_{41m}{w}_{42m}+p_{44}(w_{13m}\\ +w_{23m})\\ \\ -p_{12}{w}_{12m}+w_{14m}+p_{13}{w}_{15m}+w_{12m}{w}_{15m}\\ -w_{11m}{w}_{13m}-w_{12m}{w}_{14m}+p_{33}{w}_{41s}-p_{33}(w_{11m}\\ +w_{21m})-P_{22}{w}_{24m}+p_{23}{w}_{25m}+w_{22m}{w}_{25m}\\ -w_{21m}{w}_{23m}-w_{22m}{w}_{24m}-p_{45}{w}_{32s}-p_{45}(w_{12m}\\ +w_{22m})-p_{33}{E}_{11}-E_{14}+p_{45}{E}_{12}\end{array}\right]\end{array}$$

$$\begin{array}{c}\hfill {\theta }_1+{\theta }_2=\left[\begin{array}{c}{\theta }_{11}+{\theta }_{21}\\ {\theta }_{12}+{\theta }_{22}\\ {\theta }_{13}+{\theta }_{23}\\ {\theta }_{14}+{\theta }_{24}\end{array}\right]=\left[\begin{array}{c}-q_{11}{z}_{11m}+q_{11}{z}_{13m}+z_{42m}+z_{12m}+z_{22m}-q_{21}{z}_{21m}\\ +q_{21}{z}_{23m}-q_{41}{z}_{31m}-q_{42}{z}_{34m}{z}_{42m}{z}_{43m}+q_{41}(z_{11m}\\ +z_{21m})-b_{42}(z_{14m}+z_{15m}+z_{24m}+z_{25m})-E_{22}+q_{42}{E}_{24}\\ \\ q_{11}{z}_{12m}-q_{11}{z}_{14m}-q_{31}{z}_{41s}-q_{33}{z}_{42s}+q_{34}{z}_{43s}\\ +q_{32}{z}_{33s}{z}_{31s}+(q_{31}+q_{32}{z}_{33s})z_{31s}^3+q_{31}(z_{11m}+z_{21m})\\ -q_{33}(z_{12m}+z_{22m})-q_{34}(z_{13m}+z_{23m})+q_{31}{E}_{21}-q_{34}{E}_{23}\\ +q_{21}{z}_{22m}-q_{21}{z}_{24m}-q_{43}{z}_{31s}-q_{41}{z}_{32s}-q_{42}{z}_{34s}-z_{41s}{z}_{43s}\\ +q_{43}(z_{11m}+z_{21m})-q_{41}(z_{12m}+z_{22m})-q_{42}(z_{14m}\\ +z_{15m}+z_{24m}+z_{25m})+q_{43}{E}_{21}+q_{42}{E}_{24}\\ \\ (q_{12}-q_{11})z_{11m}-q_{11}{z}_{11m}{z}_{15m}+z_{44m}+(z_{14m}+z_{15m}\\ +z_{24m}+z_{25m})+(q_{23}-q_{21})z_{21m}+q_{23}{z}_{23m}\\ -z_{21m}{z}_{25m}+z_{41s}{z}_{42s}-E_{23}-E_{24}\\ \\ -(q_{12}-q_{11})z_{12m}+q_{13}{z}_{15m}+q_{11}{z}_{12m}{z}_{15m}-z_{11m}{z}_{13m}\\ -z_{12m}{z}_{15m}+q_{31}{z}_{41s}-q_{35}{z}_{43s}+q_{36}{z}_{44s}+q_{36}{z}_{44s}\\ +q_{36}{z}_{33s}{z}_{33s}{z}_{34s}-q_{37}(z_{11m}+z_{21m})+q_{35}(z_{13m}\\ +z_{23m})+q_{36}(z_{14m}+z_{15m}+z_{24m}+z_{25m})-(q_{23}\\ -q_{21})z_{22m}-q_{23}{z}_{24m}+z_{22m}{z}_{25m}-z_{21m}{z}_{23m}\\ -z_{22m}{z}_{24m}+q_{44}{z}_{32s}+q_{44}(z_{12m}+z_{22m})\\ -q_{37}{E}_{21}+q_{35}{E}_{23}-(q_{36}+1)E_{24}-q_{44}{E}_{22}\end{array}\right]\end{array}$$

The error systems are formulated as:

$$\begin{array}{c}\hfill {\displaystyle \frac{d^q{E}_1}{d{t}^q}}=(p_3+M_1+p_4+M_2)E_1\\ \hfill {\displaystyle \frac{d^q{E}_2}{d{t}^q}}=(s_3+M_3+s_4+M_4)E_2.\end{array}$$

(27)

Theorem 1 is confirmed if gain matrices are taken as:

$$M_1=\left[\begin{array}{cccc}0& -p_{31}& 0& -1\\ -p_{32}& 0& 0& 0\\ 0& 0& 0& 0\\ -p_{33}& 0& 0& -1\end{array}\right],M_2=\left[\begin{array}{cccc}0& -p_{42}& 0& 0\\ -p_{42}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& p_{45}& 0& 0\end{array}\right]$$

$$M_3=\left[\begin{array}{cccc}0& -1& 0& 0\\ q_{31}& 0& -b_{34}& 0\\ 0& 0& -1& -1\\ -q_{37}& 0& q_{35}& -q_{36}-1\end{array}\right],M_4=\left[\begin{array}{cccc}0& 0& 0& q_{42}\\ q_{43}& 0& 0& q_{42}\\ 0& 0& 0& 0\\ 0& -q_{44}& 0& 0\end{array}\right].$$

5. Numerical Simulation

In this section, a numerical simulation is presented to confirm the sufficiency and effectiveness of the proposed DCCHS scheme. Initial conditions of master systems (16)–(19) and corresponding slave systems (20)–(23) are as follows $(w_{11m},w_{12m},w_{13m},w_{14m},w_{15m})=(2,3,5,6,9)$; $(z_{11m},z_{12m},z_{13m},z_{14m},z_{15m})=(8,7,6,8,7)$; $(w_{21m},w_{22m},w_{23m},w_{24m},w_{25m})=(1,2,3,4,5)$; $(z_{21m},z_{22m},z_{23m},z_{24m},z_{25m})=(1,2,3,4,5)$; $(w_{31s},w_{32s},w_{33s},w_{34s})=(10,40,2.5$, $4)$; $(z_{31s},z_{32s},z_{33s},z_{34s})=(0.1,-0.5,0.1,-0.5)$; $(w_{41s},w_{42s},w_{43s},w_{44s})=(5.5,-1.25,8.4$, $2.75)$; and $(z_{41s},z_{42s},z_{43s},z_{44s})=(3.5,1.7,-4.5,2.8)$. Therefore, according to the definition of the considered DCCHS error system, the initial conditions for the error system would be $(E_{11},E_{12},E_{13},E_{14})=(3.5,5.7,3.4,30.7)$ and $(E_{21},E_{22},E_{23},E_{24})=(-5.4,10.2,-13.4,26.3)$. The phase diagrams of the FO complex systems and the FO hyper-chaotic systems for $q=0.95$ in 3-D are depicted in Figure 1a–h. The DCCHS between signals ($w_{31s}+w_{41s}$) and $(w_{11m}+w_{21m})$, signals ($w_{32s}+w_{42s}$) and $(w_{12m}+w_{22m})$, signals ($w_{33s}+w_{43s}$) and $(w_{13m}+w_{23m})$, signals ($w_{34s}+w_{44s}$) and $(w_{14m}+w_{24m})$, signals ($z_{31s}+z_{41s}$) and $(z_{11m}+z_{21m})$, signals ($z_{32s}+z_{42s}$) and $(z_{12m}+z_{22m})$, signals ($z_{33s}+z_{43s}$) and $(z_{13m}+z_{23m})$, and signals ($z_{34s}+z_{44s}$) and $(z_{14m}+z_{24m})$ are shown in Figure 2a–h. Moreover, the error of the DCCHS $(E_{11}\left(t\right),E_{12}\left(t\right),E_{13}\left(t\right),E_{14}\left(t\right),E_{21}\left(t\right),E_{22}\left(t\right)$, $E_{23}\left(t\right),E_{24}\left(t\right))$ is converging to zero, as shown in Figure 3.

A Comparative Analysis

The authors [36] used an active control for the dual combination–combination anti-synchronization, where it is observed that the synchronization error is converging to zero at $t=4.5$ (approx). In addition, the authors [31] discussed the phase CCS scheme among FO chaotic systems. It is remarked that the synchronization error state is achieved at $t=5$ (approx). Furthermore, the authors [32] discussed the dual multi-switching CCS technique. It is seen that the error state is accomplished at $t=4$ (approx). In addition, the authors [33] proposed a dual synchronization of FO chaotic systems via a linear controller, in which the synchronization error converges to zero at $t=30$ (approx). Moreover, the authors [34] studied parameter identification and the finite-time C–C synchronization and observed that the synchronization error converges to zero at $t=3$ (approx). In our current study, the DCCHS error was achieved at $t=2.5$ (approx), as exhibited in Figure 3. Hence, the synchronization time via our studied methodology is the lowest amongst all the above-discussed approaches, as shown in

Table 1. A Comparison Analysis of the current paper with previously published works.

Types of Synchronization	Time
Dual C–C anti synchronization of eight FO chaotic systems [36]	4.5
C–C phase synchronization among FO chaotic systems [31]	5
Dual C–C multi switching synchronization [32]	4
Dual synchronization of FO chaotic systems via linear controller [33]	30
Parameter Identification and Finite-Time C–C Synchronization [34]	3
Dual C–C hybrid synchronization in FO chaotic systems [Current paper]	2.5

.

6. Conclusions

In this paper, a dual combination–combination hybrid synchronization (DCCHS) scheme has been investigated in eight non-identical fractional-order chaotic/hyperchaotic systems with distinct dimensions using an active control methodology. The considered synchronization technique is attained between fractional complex chaotic systems and fractional hyperchaotic systems using the scaling matrix. It is based on the stability analysis of the fractional derivative of a linear system. With the evolution of time, the error system converges to zero asymptotically by using an appropriate and simplified active controller. The proposed DCCHS scheme has many benefits as it can give excellent protection in secure communication. Moreover, we understand that the considered approach may be generalized by utilizing numerous other control and synchronization techniques.