Complex Generalized Synchronization and Parameter Identification of Nonidentical Nonlinear Complex Systems

In this paper, generalized synchronization (GS) is extended from real space to complex space, resulting in a new synchronization scheme, complex generalized synchronization (CGS). Based on Lyapunov stability theory, an adaptive controller and parameter update laws are designed to realize CGS and parameter identification of two nonidentical chaotic (hyperchaotic) complex systems with respect to a given complex map vector. This scheme is applied to synchronize a memristor-based hyperchaotic complex Lü system and a memristor-based chaotic complex Lorenz system, a chaotic complex Chen system and a memristor-based chaotic complex Lorenz system, as well as a memristor-based hyperchaotic complex Lü system and a chaotic complex Lü system with fully unknown parameters. The corresponding numerical simulations illustrate the feasibility and effectiveness of the proposed scheme.


Introduction
Since Fowler et al. proposed a complex Lorenz system in 1982 [1], modeling, analyses and synchronization of complex systems have attracted more and more attention in nonlinear science and technology fields, the reasons of which can be roughly summed up in the following two aspects. On the one hand, some physical systems and phenomena should be accurately modeled by complex systems, such as rotating fluids, detuned lasers, disk dynamos, electronic circuits, and so on [1][2][3][4]; on the other, due to the existence of complex variables which can double the number of variables, complex systems can generate complicated dynamical behaviors with strong unpredictability, and synchronization of complex systems has widely potential applications to many fields of physics, ecological systems, signal and information processing, and system identification, especially to secure communication for achieving higher transmission efficiency and anti-attack ability [5][6][7].
As we well know, chaos synchronization is the precondition of chaotic secure communication, digital cryptography, chaotic image encryption, etc. Since the pioneering work by Pecora and Carrol in 1990 [8], chaos synchronization of real systems has been extensively investigated theoretically and experimentally, while the synchronization of complex systems has been explored for less than a decade. In the beginning stages, some synchronization schemes were directly used to synchronize complex systems, such as complete synchronization (CS) [9][10], lag synchronization (LS) [7,11], projective synchronization (PS) [12][13], phase synchronization (PhS) [14], combination synchronization [15], etc. Recently, some complex synchronization methods were presented based on their real versions. Liu et al. proposed a complex modified hybrid projective synchronization (CMHPS) scheme to synchronize complex Dadras systems, with different dimensions and complex parameters, up to a desired complex transformation matrix [16]. Wang et al. investigated a hybrid synchronization method containing complex modified projective synchronization and module-phase synchronization [17]. Sun et al. realized complex combination synchronization of three identical chaotic complex systems with complex scaling matrices [18]. Jiang et al. designed a general controller to achieve combination complex synchronization for fractional-order chaotic complex systems [19]. It is worth noting that Refs [20][21] have explored the synchronization issues of complex systems with unknown parameters which are likely to exist in practice. Zhang et al. investigated the complex modified projective synchronization (CMPS) and parameter identification of uncertain real chaotic systems and complex chaotic systems [20]. Liu et al. used an adaptive complex modified projective synchronization (ACMPS) method to synchronize two chaotic (hyperchaotic) complex systems up to a complex scaling matrix, and to estimate the unknown complex parameters successfully [21].
Based on the above-mentioned complex synchronization methods, the response complex systems can be synchronized with the drive complex systems up to the desired complex scaling matrices. Shall we further generalize these synchronization schemes and synchronize the complex systems with respect to a given complex functional relationship? That is, can generalized synchronization (GS) be extended to synchronize complex systems? Rulkov et al. firstly proposed the generalized synchronization, where two chaotic systems are said to be synchronized if a given functional relation can be realized between the variables of drive and response systems [22]. With different given functions, GS can degenerate to various PSs, antisynchronization (AS) and CS. Furthermore, the given functions are almost impossible to be predicted, which can enhance secure performance when GS is applied to chaotic secure communication.
In the recent two decades, GS of chaotic or hyperchaotic real systems has been widely investigated. For instance, Refs [23][24][25] realized GS of different chaotic and hyperchaotic systems, while Refs [26][27][28] achieved adaptive generalized synchronization (AGS) and parameter identification of different chaotic systems with unknown parameters. However, to our best knowledge, up to now, there are few published achievements on CGS of nonidentical nonlinear complex systems. So, it is meaningful and challenging to extend GS from real systems to complex systems, and to realize CGS and parameter identification of chaotic and hyperchaotic complex systems with unknown parameters.
Motivated by the above discussions, this paper investigates CGS and parameter identification of different chaotic and hyperchaotic complex systems with unknown parameters. In practice, the parameters of some nonlinear systems cannot be exactly known, so we choose uncertain nonlinear complex systems as the research objects, and use adaptive control and Lyapunov stability theory to design CGS and parameter estimation scheme for them. In our proposed scheme, CGS is defined by extending GS from real space to complex space, and designed with consideration of error feedback control gains which are introduced to adjust converging velocity. Furthermore, according to the orders of the drive and response nonlinear complex systems (i.e., same-order, increased-order, and reduced-order), three different examples are presented to verify the correctness, feasibility, and efficiency of the proposed scheme.
The rest of this paper is organized as follows. The definition and design of CGS of nonidentical complex systems are given in Section 2. CGS and parameter identification of a memristor-based hyperchaotic complex Lü system and a memristor-based chaotic complex Lorenz system with the same orders, a chaotic complex Chen system and a memristor-based chaotic complex Lorenz system via increased order, as well as a memristor-based hyperchaotic complex Lü system and a chaotic complex Lü system via reduced order, are investigated theoretically and illustrated numerically in Section 3-5, respectively. Finally, some conclusions are drawn in Section 6.

Definition of CGS
Consider the following nonidentical drive and response complex systems with fully unknown parameters where x = (x 1 , x 2 , Á Á Áx n ) T and y = (y 1 , y 2 , Á Á Áx m ) T are complex state vectors of the drive system (1) and response system (2) respectively, x k = x k,r + jx k,i (k = 1, Á Á Á, n), y k = y k,r + jy k,i (k = 1, Á Á Á, m), j ¼ ffiffiffiffiffiffi ffi À1 p , the subscripts r and i denote the real and image parts of the complex variables, vectors and matrices throughout this paper. θ2R p and δ2R q are real vectors of unknown parameters. F(x)2C n×p and G(y)2C m×q are complex matrices, Remark 1 Some nonlinear complex systems can be formed as system (1), such as complex Lorenz system, complex Chen system, complex Lü system, memristor-based complex Lorenz system, memristor-based complex Lü system, and so on. For synchronizing such complex systems, the complex variables and functions could be divided into the real parts and imaginary parts.

General scheme of CGS and parameter identification
Define the complex CGS error vector as where e = (e 1 , e 2 , Á Á Áe m ) T 2C m , e r = (e 1,r , e 2,r , Á Á Áe m,r ) T 2R m ,e i = (e 1,i , e 2,i , Á Á Áe m,i ) T 2R m . By taking the derivative of Eq 4 with respect time, the CGS error dynamical system is obtained as where J(ϕ)2C m×n is the Jacobian matrix of ϕ(x), and J(ϕ) = J r (ϕ) + jJ i (ϕ). By substituting Eqs 1 and 2 into Eq 5, Eq 5 can be represented by Therefore, the problem of CGS for two nonidentical complex systems (1) and (2) is transformed to the stability analysis of zero solution of the error dynamical system (6). Adaptive CGS scheme is given in Theorem 1 and is proved based on Lyapunov stability theory.
Proof We introduce a positive Lyapunov function as The time derivative of V(t) along the trajectories of the error dynamical system (6) is calculated as Substituting Eqs 7 and 8 into Eq 10, then Based on Lyapunov stability theory, since V(t) and _ V ðtÞ are positive and negative respectively, the CGS errors and the parameter errors asymptotically converge to zero as the time tends to infinity, i.e., lim t!1 e r ðtÞ ¼ 0,lim t!1 e i ðtÞ ¼ 0, lim t!1ỹ ðtÞ ¼ 0 and lim t!1d ðtÞ ¼ 0, which indicate that CGS and parameter identification are realized. The proof is completed.

CGS of a Chaotic Complex Chen System and a Memristor-Based Chaotic Complex Lorenz System (n<m)
In this section, we investigate CGS of two nonidentical complex systems via increased order. A chaotic complex Chen system, investigated in [9], is introduced as the complex drive system, which is described as where x 1 , x 2 2C, x 3 2R, c 1 , c 2 and c 3 are unknown real parameters. When c 1 = 27, c 2 = 23, c 3 = 1, and x(0) = [−3 − 2j, −1 − 5j, −4] T , the complex Chen system (19) operates in chaotic orbits, as shown in Fig 6. The memristor-based chaotic complex Lorenz system, i.e., system (13), is also served as the complex response system.
Numerical simulations are presented to verify the validity and effectiveness of CGS between systems (19) and (13)

Conclusions
This paper investigates a novel synchronization scheme named complex generalized synchronization, and its application to synchronization and parameter identification of two nonidentical complex nonlinear systems with fully unknown parameters. An adaptive controller and a parameter estimator are proposed and proved theoretically based on Lyapunov stability theory.
Three illustrative examples are presented to verify the correctness and effectiveness of the proposed scheme, namely, CGS of a memristor-based hyperchaotic complex Lü system and a memristor-based chaotic complex Lorenz system, CGS of a chaotic complex Chen system and a memristor-based chaotic complex Lorenz system, as well as CGS of a memristor-based hyperchaotic complex Lü system and a chaotic complex Lü system. The proposed CGS scheme has some advantages, for instance, it can be applied to synchronize complex systems with different orders (generalizability), can be transformed to other types of synchronization with different given complex map vectors (feasibility), can be achieved in a short time with the appropriate control strength (timelines), and can be almost impossibly predicted with the complex map vector (security). So, CGS has extensively potential applications to secure communication, digital cryptography, and so on, which will be involved in our future works.

Author Contributions
Conceived and designed the experiments: XW SW. Performed the experiments: SW BH. Analyzed the data: SW XW BH. Contributed reagents/materials/analysis tools: SW BH. Wrote the paper: SW.