Analytical and Numerical Study of the Projective Synchronization of the Chaotic Complex Nonlinear Systems with Uncertain Parameters and Its Applications in Secure Communication

The main aim of this research is to find an analytical and numerical study to investigate the projective synchronization of two identical or nonidentical chaotic complex nonlinear systems with uncertain parameters.The secure communication between these systems is achieved based on this study. Based on the adaptive control technique and the Lyapunov function a scheme is designed to achieve projective synchronization of chaotic attractors of these systems. The projective synchronization of two identical complex Chen systems and two different chaotic complex Lü and Lorenz systems is taken as two examples to verify the feasibility of the presented scheme.These chaotic complex systems appear in several applications in physics, engineering, and other applied sciences. Numerical simulations are calculated to demonstrate the effectiveness of the proposed synchronization scheme and verify the theoretical results. The above results will provide theoretical foundation for the secure communication applications based on the proposed scheme.


Introduction
In 1999, projective synchronization has been first reported by Mainieri and Rehacek [1] in partially linear systems that the drive and response systems synchronize up to a constant scaling factor . Later Xu and Li showed that projective synchronization could be extended to general classes of chaotic systems without partial linearity [2].Complete synchronization and antisynchronization are the special cases of the projective synchronization where the scaling factor  = 1 and  = −1, respectively.
In applied sciences and engineering there are a lot of problems involving complex variables which are described by these complex systems, for example, when amplitudes of electromagnetic fields and atomic polarization are involved.Increasing the number of variables (or introducing complex variables) is also crucial in chaos synchronization used in secure communications, where one wishes to maximize the content and security of the transmitted information.Increasing the number of variables and parameters in studying projective synchronization of chaotic complex systems is of course crucial in the area of secure communication where one wishes to maximize the content and security of the transmitted information [19,23].Secure communication means that two entities are communicating with each other in a way that does not allow anyone else to understand their message.So, we hope to achieve the secure communication based on the proposed scheme of projective synchronization of chaotic complex systems.
A dynamical system is called chaotic if it is deterministic, has a long-term periodic behavior, and exhibits sensitive dependence on the initial conditions.If the system has one positive Lyapunov exponent then the system is called chaotic [27].
Consider the -dimensional chaotic complex nonlinear system as follows: where x = ( 1 ,  2 , . . .,   )  is a state complex vector, x = x  + x  , x  = ( 1 ,  3 , . . .,  2−1 )  , x  = ( 2 ,  4 , . . .,  2 )  ,  = √ −1,  denotes transpose, F(x) is  ×  complex matrix and the elements of this matrix are state complex variables, A is  × 1 complex (or real) vector of system parameters, f = ( 1 ,  2 , . . .,   )  is a vector of nonlinear complex functions, and superscripts  and  stand for the real and imaginary parts of the state complex vector x.The purpose of this paper is to investigate the phenomenon of the adaptive projective synchronization of two identical or different systems of the form (1) with fully unknown parameters by designing an adaptive control scheme.
Most of chaotic complex systems can be described by (1), such as complex Lorenz, Chen, and Lü systems [14,19].In order to show the results of our scheme of two identical or nonidentical systems of the form (1) we choose, as an example, the chaotic complex Chen, Lorenz, and Lü systems which have been introduced and studied recently in our works [16,19].
The organization of this paper is as follows.Design of the proposed scheme for adaptive projective synchronization of two identical or different -dimensional chaotic complex nonlinear systems with fully unknown parameters is stated in Section 2. In Section 3 we study projective synchronization of two identical chaotic complex Chen systems as an example for Section 2, while we investigate projective synchronization between the chaotic complex Lorenz system and the chaotic complex Lü system in Section 4. The secure communication based on the results of projective synchronization of two chaotic complex Chen systems is shown in Section 5. Finally, the main conclusions of our investigations are summarized in Section 6.

A Scheme for Adaptive Projective Synchronization
We consider two different -dimensional chaotic complex nonlinear systems of the form (1); one is the master system as and the second is the controlled slave system as where the additive complex controller The adaptive synchronization problem is to design a controller L, estimate the unknown parameters of the master and slave systems, and make the slave system follow the master system and become ultimately the same.

Theorem 1. One may be able to achieve the adaptive projective synchronization between systems (5) and (6) by a choice of the controller L as
and the adaptive laws of parameters are selected as where e() = y  − x  = e  + e  = ( 1 ,  2 , . . .,   )  is the vector of the complex error function Proof.We subtract ( 5) from ( 6) to get Therefore, we will use Lyapunov function as The total time derivative of () along the trajectory of the error system ( 9) is as follows: where Ȧ = − Ȧ and Ḃ = − Ḃ.
Consequently, the states of the slave system and the master system will be globally synchronized asymptotically.This completes the proof.
Remark 2. If systems ( 5) and ( 6) satisfy f(⋅) = g(⋅) and F(⋅) = G(⋅), then the structure of system (5) and system (6) is identical.Therefore, our scheme is also applicable to the adaptive synchronization of two identical chaotic systems with fully unknown parameters and the adaptive laws of parameters are selected as Finally, our scheme is illustrated by applying it for two identical Chen systems in Section 3 and two different chaotic complex Lorenz and Lü systems in Section 4.

Projective Synchronization between Two
Complex Chen Systems 3.1.Analytical Formula of Controller.Let us now investigate the projective synchronization of two identical chaotic complex Chen systems with uncertain parameters as an example for Section 2. The master and the slave systems are thus defined, respectively, as follows: where complex and real control functions, respectively, which are to be determined.The complex systems ( 14) and ( 15) can be formed, respectively, as ) . ( So, by comparing the complex systems (16) with the form of systems ( 5) and ( 6), respectively, we find ) . ( According to Theorem 1, the controller is designed as where    =   −   ;  = 1, 2, 3, 4, 5, 7. Since A = B = (, , )  we can calculate the adaptive laws of parameters by using (13) as ) . (19)

Numerical Results.
To verify and demonstrate the feasibility of the proposed scheme, we discuss the simulation results of the projective synchronization between two identical chaotic complex Chen systems ( 14) and (15).Systems ( 14) and ( 15) with the controller (18) are solved numerically, and the parameters are chosen as  = 42,  = 4, and  = 26.
In Figure 1 the solutions of ( 14) and ( 15) are plotted subject to different initial conditions and show that projective synchronization is indeed achieved.In Figure 1(a) we select  = −1 and the attractors in ( 1 ,  3 ,  5 ) space of master system (14) and slave system (15) have the same size but opposite shape.But when we choose  = −2 in Figure 1(b) the attractors of ( 14) and ( 15) have the opposite shape in ( 1 ,  3 ,  5 ) space, but the size of the attractor of the slave system is twice as big as of the master system.Figure 2 shows that the estimated values of the unknown parameters â(), ĉ(), and b() converge to 42, 26, and 4, respectively.These results ensure that our scheme is suitable for effecting adaptive projective synchronization of two identical chaotic complex nonlinear systems.

Projective Synchronization between Complex Lorenz and Lü Systems
4.1.Analytical Formula of Controller.This subsection is devoted to test the validity of the Scheme of Section 2 by applying it to the chaotic complex Lorenz as a master system and Lü complex as a salve system with fully unknown parameters.

The Application in Secure Communications
In this section, secure communications scheme based on projective synchronization between two identical chaotic complex Chen systems is investigated.We consider the two chaotic complex Chen systems as transmitter and receiver systems.The message signal () and chaotic signals of the transmitter system are encrypted by means of an invertible nonlinear function Ξ = (, 1 ,  2 ,  3 ,  4 ,  5 ).
Then we add the signal Ξ to one of the five variables  1 ,  2 ,  3 ,  4 ,  5 ; for instance, we inject it into the variable  3 so the combined signal is Δ() = Ξ +  3 .Then, chaotic signals of the transmitter system and combined signal are transmitted to the receiver side.In the receiver, the controller L can be constructed by (18), so the projective synchronization between two chaotic complex Chen systems will be achieved after some time and the states of X will approach Y/ where  is a nonzero constant scaling factor and increases the content and security of the transmitted message.At a certain time the receiver starts to recover Ξ through a simple transformation Ξ = Δ() −  3 /.Finally, since the nonlinear function  is invertible, the message signal can be recovered as ȓ () =  −1 ( 1 ,  2 ,  3 ,  4 ,  5 , Ξ).In the following numerical simulations the system parameters, initial conditions of the transmitter, and receiver systems are chosen as the same values as those in Section 3 and the constant scaling factor  = 0.7.We choose the invertible function as Ξ =  1 + arctan(()); () = cos(2) and we assume that the signal Ξ is added to the variable  3 .The numerical simulation for the application of projective synchronization in secure communication is Figure 5(c) displays the recovered message ȓ ().The error between the original message and the recovered one is shown in Figure 5(d).From Figure 4(d), it is easy to find that the information signal () is recovered accurately after a short transient.

Conclusion
Synchronization and control are important topics which have been studied to date primarily on dynamical systems described by real variables in applied nonlinear sciences.There also exist, however, interesting cases of dynamical systems where the main variables participating in the dynamics and  is a constant scaling factor.The elements of the vectors Ã and B are the parameters estimations of elements of the vectors A and B, respectively; the parameters errors are defined as Â = A − Ã and B = B − B and Ψ = diag( 1 ,  2 , . . .,   ) and  = diag( 1 ,  2 , . . .,   );   and   are positive constants;  = 1, 2, . . ., .

= 4 ,
and μ(0) = 5.The results of adaptive projective synchronization of two different chaotic complex Lorenz and Lü systems are shown in Figure3.In Figures3(a) and 3(b) we plotted hyperchaotic attractors for different values of  as  = −1 and −2, respectively.It is clear that, from Figure 3(a), the attractors in ( 1 ,  3 ,  5