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Depending on double compound synchronization and compound combination synchronization, a new kind of synchronization is introduced which is the double compound combination synchronization (DCCS) of eight n-dimensional chaotic systems. This kind may be considered as a generalization of many types of synchronization. In the communication, based on many of drive and response systems, the transmitted and received signals will be more secure. Using the Lyapunov stability theory and nonlinear feedback control, analytical formulas of control functions are obtained to insure our results. The corresponding analytical expression and numerical treatment are used to show the validity and feasibility of our proposed synchronization scheme. The eight memristor-based Chua oscillators are considered as an example. Other examples can be similarly investigated. The proposed synchronization technique is supported using the MATLAB simulation outcomes. We obtain the same results of numerical treatment of our synchronization using simulation observations of our example.
Chaos synchronization and control theory have a great interest due to their potential applications in several fields during the recent decades.[1–5] Different synchronization types have been introduced and many theoretical results have been achieved, such as complete synchronization,[6–8] anti-synchronization,[9–11] lag synchronization,[12,13] projective synchronization,[13,14] function projective synchronization,[15] etc. Most of the pervious types of synchronization concentrate on synchronization between one drive and one response systems. There exist many techniques of control to achieve synchronization such as sliding mode control, random phase control, adaptive passive control, and others.[16–24]
Synchronization of systems with more than one drive and response systems is very interesting and important, because the transmitted signals may own stronger anti-translated and anti-attack ability. Complexity and the formation of the drive signals are also peppy factors to include communication security. Researchers studied complex schemes to transmit the information signals. Runzi et al. investigated combination synchronization of two drive and one response systems,[25,26] while the synchronization of two drive and two response systems (combination–combination synchronization) is studied by Sun et al.[27] Mahmoud et al. introduced the generalization of combination–combination synchronization of chaotic systems.[28,29] A new kind of compound synchronization (synchronization between three drive systems and one response system) has been illustrated in Ref. [30]. Zhang and Deng studied the double compound synchronization, which is the synchronization of four drive and two response systems.[31] The compound–combination synchronization (synchronization of three drive and two response systems) was presented in Ref. [32].
In this paper, we introduce a new type of synchronization which is a generalization of many pervious types of synchronization and more complexity. This type is called double compound combination synchronization (DCCS) among eight chaotic systems (four drive and other four response chaotic or hyperchaotic systems). In the last years, a new circuit element, namely memristor which is basically a fourth class of electrical circuit, joining the inductor, the capacitor, and the resistor. Memristor produced the circuit systems which have chaotic and hyperchaotic property. In Ref. [33], the authors replaced Chua’s diodes with memristors which proposed several nonlinear oscillators from Chua’s oscillators. Wen et al. regarded the fuzzy modeling and synchronization of memristor-based Lorenz circuits with memristor-based Chua’s circuits.[34] Memristor was postulated as the missing fourth passive circuit element.[35] A great breakthrough that a passive two terminal physical implementation was immediately related to memristor theory was obtained by Hewlett–Packard Labs.[36] This new circuit element can be useful for low-power computation and storage to store information and data.[37] Memristor, also, can be used to implement programmable analog circuits, leveraging memristor’s fine-resolution programmable resistance without causing perturbations due to parasitic components,[38] and soon.
This paper is organized as follows: Section
We will present the technique which is used to achieve the DCCS of four drive and four response n-dimensional chaotic systems. The first scaling drive system is given as follows:
![]() | Fig. 1. (color online) The topological structure diagram of the drive systems ( |
Let Ai = diag (ai1, ai2, …, ain), Bi = diag (bi1, bi2, …, bin), i = 1, 2, 3, 4, therefore, the error system can be described as:
In this section, we apply the synchronization scheme of Section
We designed circuit implementation for system (
![]() | Fig. 4. (color online) Simulation observations of system ( |
The second scaling drive memristor-based Chua oscillator is given by
According to Theorem
From Eq. (
In this section, numerical treatments are illustrated to state the effectiveness of the suggested scheme of DCCS of eight identical memristor-based Chua oscillators. If we take, ci1 = di1 = 15.6, ci2 = di2 = 3, ci3 = di3 = 21, ci4 = di4 = 15, ci5 = di5 = 0.5, (i = 1, 2, 3, 4),
Synchronization results also have been calculated from MATLAB/Simulink and these results are shown in Figs.
The object of this paper is to propose a new type of synchronization among eight n-dimensional chaotic systems. The proposed scheme is used to achieve the DCCS between four drive and four response n-dimensional chaotic systems. Using the Lyapunov stability theory and nonlinear feedback control, analytical expressions are obtained for achieving DCCS. As an example, we consider the eight identical memristor-based Chua chaotic systems. Numerical results have been calculated to illustrate the validity of our scheme. Based on complexity of DCCS, our results make more secure and interesting to transmit and receive signals in applications of communications. The numerical results of DCCS are shown in Figs.
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