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

: 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.


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], • The proposed DCCHS methodology considers eight non-identical, complex, FO chaotic/ hyperchaotic systems. • It describes a robust DCCHS scheme-based controller to achieve dual combinationcombination 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.
is a constant function and of the order q > 0, the Caputo fractional-order derivative satisfies the conditions: Property 2. The Caputo derivative satisfies the following linear property: where Ψ 1 (t) and Ψ 2 (t) are functions of t, and c 1 and c 2 are constants.

Problem Formulation
The authors [23] introduce the scheme of dual combination-combination hybrid synchronization (DCCHS). Firstly, two master systems are taken as: Consider the next two master systems as: For the master systems, there are two slave systems which are taken as: D q z 3s = s 3 z 3s + g 3 (z 3s ) + θ 1 .
Define the error state functions as follows: The error dynamics system turns out to be: Theorem 1 ([23,32]). If the control functions of the slave systems are chosen in the following manner: where M 1 , M 2 ∈ R n 2 ×n 2 are obtained from p 3 and p 4 , respectively, also called gain matrices, and M 3 , M 4 ∈ R n 2 ×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 (13), substitute the control function values φ and θ from Equation (14). The error dynamics system (13) has been reduced to the following form:

Proof. In Equation
Using Definition 2, the error system (15) becomes stable asymptotically using active control if and only if each eigenvalue λ i of (p 3 Consequently, lim t→∞ E = 0, and therefore, the discussed systems attained the DCCHS scheme.

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: 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: The mathematical model of FO complex T system [39] can be described as: 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: The mathematical model of the FO complex Lu system [40] is presented as: 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.
The error function will be written as: Matrices ρ 1 and ρ 2 are chosen as: Finally, the error function can be obtained as: where Using Theorem 1, the controllers would be: The error systems are formulated as: Theorem 1 is confirmed if gain matrices are taken as:

A Comparative Analysis
The authors [36] used an active control for the dual combination-combination antisynchronization, 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 abovediscussed approaches, as shown in Table 1. 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

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