Generalized function projective synchronization of chaotic systems for secure communication

By using the generalized function projective synchronization (GFPS) method, in this paper, a new scheme for secure information transmission is proposed. The Liu system is employed to encrypt the information signal. In the transmitter, the original information signal is modulated into the system parameter of the chaotic systems. In the receiver, we assume that the parameter of receiver system is uncertain. Based on the Lyapunov stability theory, the controllers and corresponding parameter update rule are constructed to achieve GFPS between the transmitter and receiver system with uncertain parameters, and identify unknown parameters. The original information signal can be recovered successfully through some simple operations by the estimated parameter. Furthermore, by means of the proposed method, the original information signal can be extracted accurately in the presence of additional noise in communication channel. Numerical results have verified the effectiveness and feasibility of presented method.


Introduction
Chaos is a kind of characteristics for nonlinear systems, which is a bounded unstable dynamic behavior that exhibits sensitive dependence on initial conditions and includes infinite unstable periodic motions. Since Pecora and Carroll [1] presented the conception of chaotic synchronization for two identical chaotic systems with different initial conditions, many synchronization methods have been proposed, such as complete synchronization (CS) [1], generalized synchronization [2], phase synchronization [3], impulse synchronization [4], lag synchronization [5], projective synchronization [6][7][8], etc. Amongst all kinds of chaos synchronization, projective synchronization, first reported by Mainieri and Rehacek [6], has been especially extensively studied because it can obtain faster communication with its proportional feature [9][10][11][12].
However, the above projective synchronization (PS) method is characterized that its drive and response systems are synchronized up to a constant scaling factor. Recently, Chen et al. [13] introduced a new PS scheme which is called function projective synchronization (FPS), where the responses of synchronized dynamical states can synchronize up to a scaling function factor. Let the scaling function be constant or unity, one can obtain PS or CS. So FPS is a more general definition of PS. Be-cause the unpredictability of the scaling function in FPS can additionally enhance security of communication, this feature could be applied to get more secure communication. More recently, many studies concentrate on FPS of chaotic systems and its application to secure communication [13][14][15][16][17][18][19][20]. For instance, FPS of two identical or different chaotic systems was studied in [13,14,16,18]. In [15,20], another new synchronization phenomenon, generalized function projective synchronization (GFPS), was proposed, in which drive and response systems could be synchronized to a scaling function matrix. In [20], Yu and Li investigated GFPS of two entirely different systems with fully unknown parameters. Du et al. [15] studied GFPS in coupled chaotic systems and its application in secure communication.
In the past decades, the use of chaotic signals for information transmission attracts great attention of modern scientists from various fields [16,17,[20][21][22][23][24][25][26]. Different approaches for transmission of information signals using chaotic dynamics have been proposed, such as chaotic masking, chaotic modulation, nonlinear mixing, chaotic switching, and others. In a typical chaotic synchronization communication scheme, the information to be transmitted is carried from the transmitter to the receiver by a chaotic signal through an analog channel. In the receiver, chaos synchronization is employed to recover the information signal. In many existing secure communication methods, the information signal is directly added to input of chaotic systems. The magnitude of transmitted signals is required to be sufficiently small, otherwise it may lead to the instability of whole system. On the other hand, although these communication methods have been successfully demonstrated in simulations, performance of communication schemes were usually quantified by assuming the identical chaos synchronization based on exact knowledge of the system parameters [27,28], which may impose some limitations to applicability of these techniques. But in real situation, some or all of parameters are unknown and the noise exists. The effect of these uncertainties and noise will destroy the synchronization and even break it. As a result, one cannot extract the original information in the receiver. Therefore, it is essential to study secure communication in the presence of unknown parameters and noise.
Motivated by the above discussions, this paper proposes a new secure communication scheme based on GFPS of uncertain Liu chaotic system and parameter modulation. In the transmitter, the original information signal is firstly transformed by an invertible function. Then the processed signal is modulated into the parameter of Liu system. The resulting system is still chaotic. In our method, no constraint is imposed on the magnitude of the original information signal. Suppose the parameter of the receiver system is unknown. Based on the Lyapunov stability theory, we design the controllers and the parameter update rule to realize GFPS of uncertain Liu chaotic systems and identify the unknown parameter of the receiver system. Then the information signal in the receiver can be recovered by the estimated parameter. Moreover, it is worth noting that secure communication using GFPS can still be realized fast even if the transmission channel is perturbed by additive noise.
The rest of this paper is organized as follows. In Section 2, Liu system is described briefly and the definition of GFPS is presented. Section 3 gives the chaotic secure communication scheme using GFPS and parameter modulation. By means of the Lyapunov stability theory and adaptive control, the controllers and corresponding parameter update rule are designed to ensure GFPS between two identical Liu chaotic systems with uncertain parameters. Simulation experiments of the proposed secure communication system have been performed in Section 4. The conclusions are finally drawn in Section 5.
2 System description and the definition of GFPS

The Liu system
In 1963, Lorenz [29] found the first classical chaotic attractor in a three-dimensional autonomous system derived form a simplified model of earth atmospheric convection system. As the first chaotic model, the Lorenz system has become a paradigm of chaos research. Mathematicians, physicists and engineers from various fields have thoroughly studied the essence of chaos, characteristics of chaotic systems, bifurcations, routes to chaos, and many other related topics. There are also some chaotic systems of great significance that are closely related to the Lorenz system but not topologically equivalent to it, such as the Rössler system [30], the Chen system [31] and the Lü system [32]. Recently, Liu et al. [33] proposed a system of three-dimensional autonomous differential equations with only two quadratic terms, which is described as follows: where (x, y, z) R 3 are state variables, (a, b, c) R 3 are all positive real parameters and c varies in a certain range. In the Liu system, when the parameter c varies in a big range, the resulting system can still be chaotic and have more abundant and complex behaviors than the original Liu system. And compared to other parameters, the parameter c is more suitable to be used in parameter modulation. Figure  The Liu system has a butterfly-shaped attractor similar to the Lorenz attractor but not equivalent. The existing studies have shown that the Liu system is equivalent to Shimizu-Morioka system and is a representative example of chaotic attractors. Compared to other chaotic systems, when the system parameters of Liu system vary in a certain range, this chaotic system can still be chaotic, further the dynamical behaviors of this system become  more abundant and complex. Therefore, it is possible to modulate information signal into the parameter of chaotic systems to realize chaotic secure communication.
Remark 1 It is found that many chaotic systems can maintain chaotic behavior when a system parameter continuously changes in a certain range. We can obtain a "new" system parameter by modulating the information signal into the parameter of chaotic systems. If the "new" system parameter can ensure the corresponding system still be chaotic, the detector cannot extract the information signal from the signals, and transmitted in the channel. So it is possible to modulate the information signal into the parameter of chaotic systems to realize chaotic secure communication.

The definition of GFPS
The drive system and the response system are defined as follows: where x = (x 1 , x 2 , ..., x n ) R n , y = (y 1 , y 2 , ..., y n ) R n are the state vectors; f, g : R n R n are differentiable functions; U(x, y) is a controller to be designed. Let us define the error vector as e = y − Λx (4) where e = (e 1 , e 2 , ..., e n ) T , and Λ = diag(j 1 (t), j 2 t, ..., j n (t)) is reversible and differentiable, where j i (t): R n R(≠ 0) is a continuously differentiable functions with bounded.
Definition 1 (GFPS) For the drive system (2) and the response system (3), it is said that system (2) and (3) are GFPS, if there exists a scaling function matrix Λ such that lim t ∞ ||e|| = 0.

Remark 2
We call Λ a scaling function matrix and j i (t) a scaling function factor, respectively. It is easy to see that if j 1 (t) = j 2 (t) = ... = j n (t), GFPS is simplified to FPS. If j i = a i (i = 1, 2, ..., n) where a i R and a i ≠ 0, GPS will occur. If Λ = lI where l R is a constant and I n is an n × n identity matrix, GFPS is reduced to PS. In particular, if j i (t) = 1 or j i (t) = -1, the problem further becomes standard CS or anti-synchronization (AS).
3 Secure communication based on GFPS and parameter modulation 3.1 The chaotic secure communication scheme using GFPS and parameter modulation The secure communication system involves the development of a signal that contains the information which is to remain undetectable by others within a carrier signal. We can ensure the security of this information by inserting it into a chaotic signal which is transmitted to a prescribed receiver that would be able to detect and recover the information from the chaotic signal.
In the present application, we propose a new chaotic secure communication scheme using GFPS of uncertain Liu chaotic system and parameter modulation, as shown in Figure 3. The system consists of a transmitter module, a communication channel and a receiver module. In this scheme, the chaotic signal is generated by using Liu chaotic system which was described by the system model of differential equation (1). For convenience, the chaotic system is written as the following form: where the state vector X = (x, y, z). The transmitter module is composed of a chaotic system S 1 and an invertible transformation function . The "new" parameter b is formulated as a function of the original information signal f (t), i.e., b = (f (t)), which meets the need of the parameter range of c. So the resulting system still exhibits a chaotic behavior. In our method, the following two cases: no circumstance noise and additive noise in the transmission channel will be considered. The receiver module consists of another chaotic system S 2 , the controller U(S 1 , S 2 ) and the corresponding inverse function -1 . Assume that the parameterβ of the receiver system S 2 is uncertain. In the receiver, with the help of controller U(S 1 , S 2 ), GFPS between systems S 1 and S 2 can be realized. The unknown parameterβ can be asymptotically identified as GFPS appears. Then the original in-formation can be recovered through some simple operations by the function -1 and the estimated parameterβ, i.e., f (t) = ϕ −1 (β) .

GFPS of uncertain Liu chaotic system
As we all know, the information signal is often bounded; in other words, the information signal f (t) satisfies where m and M are known constants. For the Liu system, we will employ the parameter to transmit the information signal. Define a "new" parameter b as follows: Obviously, the "new" parameter satisfies b [0. 5,8]. According to the above description of Liu system, we know the following system ⎧ ⎪ ⎨ ⎪ ⎩ẋ 1 = 10(y 1 − x 1 ) will still be chaotic, which has more abundant dynamic behavior. Since the resulting system (8) is chaotic, it is hard to detect information signal from the signals transmit-ted in the channel. We take the system given by Equation (8) as the transmitter system and the receiver system is constructed as follows: where u i (t) (i = 1, 2, 3) are controllers to be designed, β is unknown parameter of receiver system which needs to be estimated. (10) and denote the parameter estimate error as
Proof We choose the following Lyapunov function for the error system (13): Taking the time derivative of V(t) along the trajectories of the error system (13) yieldṡ Substituting Equations (14) and (15) into Equation (17), one can obtaiṅ where Q = diag (10, 1, b ) and b [0. 5,8]. Clearly, Q is a positive definite matrix andV(t) is negative definite. Based on the Lyapunov stability theory, the error dynamical system (13) is globally and asymptotically stable at the origin, and we have e b 0, as t ∞ . Hence GFPS between the transmitter system (8) and the receiver system (9) is achieved and the uncertain parameter is also identified in the receiver end simultaneously under the controllers (14) and the parameter update rule (15). The proof is completed.
When GFPS between the transmitter system and the receiver system appear, ac-cording to Theorem 1, the uncertain parameterβ can be estimated asymptotically, i.e.,β → β. So the original information signal can be recovered in the receiver as wheref (t) denotes the recovered signal. Thus we havẽ which implies that the original information signal can be recovered successfully in the receiver end.
Remark 3 In practical situations, if the information signal to be transmitted is too large, it will result in a chaotic system to be asymptotically stable or emanative. In this case, one may fail to extract the information signal. In the presented method, no constraint is imposed on the information signal. The original information signal is firstly transformed by an invertible function. Then the transformed signal is used as the parameter of Liu system. The resulting system exhibits more abundant chaotic behavior. The interceptor cannot extract the information from the transmitted signals in the channel.

Secure communication without circumstance noise
In this section, we focus on considering the case of no circumstance noise in the transmission channel. We choose the original information signal as f (t) = 1.5 + 2.5 sin(2t). The scaling functions are selected arbitrarily as j 1 (t) = 0.5 sin t+2, j 2 (t) = 3 cos(2 * t) -1 and j 3 (t) = -1.5. It is obvious that -1 ≤ f (t) ≤ 4. Let θ = 2, by Equation (7), we obtain the "new" parameter as b = 0.5 sin (2t) + 2.5. Clearly, b [2,3]. Therefore, the resulting system (8) is still chaotic, as depicted in Figure 4. So one cannot extract the information signal from the transmitted signals x 1 , y 1 and z 1 . The simulation results of the proposed secure communication scheme using GFPS between systems (8) and (9) by the controllers (14) and the parameter update rule (15) are shown in Figure 5. Figure 5 plots the original information signal f (t). Figure 5 displays the time evolution of the GFPS errors, which shows that the time response of the GFPS errors e i (t)(i = 1, 2, 3) converge to zero after t > 0.4. The recovered signalf (t) is shown in Figure 5 and and the error signal between the original information signal and the recovered one is plotted in Figure 5. The information signal f (t) and the scaling functions j i (t) are chosen as those above. Set θ = 4. Then the "new" parameter b = 0.5 sin(2t) + 4.5. Obviously, b [4,5]. Figure 6 displays that the resulting system (8) exhibits chaotic behavior. The numerical results of the proposed secure communication scheme are illustrated in Figure 7. Figure 7 shows that the synchronization errors e i (t) have been stabilized at the origin after t > 0.25, which implies that GFPS between the transmitter system (8) and the receiver system (9) is achieved. The original information signal and the recovered one are depicted in Figure 7. Figure 7 displays the error signal between the original information signal and the recovered one, from which it is found that the original information signal f (t) can be extracted exactly after t > 0.25. Figure 8 describes the logarithm of the absolute value of the error signal when b = 0.5 sin(2t) + 2.5 and b = 0.5 sin(2t) + 4.5, respectively. From Figure 8, we find that the synchronization rate when b = 0.5 sin(2t) + 4.5 is faster than that when b = 0.5 sin(2t) + 2.5, and   the recovered signal when b = 0.5 sin(2t) + 4.5 is more accurate than that when b = 0.5 sin(2t) + 2.5.
Similarly, we have also tested the proposed secure communication scheme for other values of θ. Limited to the length of this paper, we omit these results here. Numeric evidence shows that the larger the value of θ, the faster to achieve GFPS, and the more accurate the recovered signal is.

Secure communication with additive noise
Here, we will study the proposed chaotic secure communication scheme when there exists additive noise in the channel. Suppose the additive noise N(t) = (0.01 sin (10t), 0.02 sin(20t), 0.03 sin(30t)). Choose the information signal as f (t) = 4 cos(0.5t) -3. The scaling functions are selected randomly as j 1 (t) = 1 + e -t , j 2 (t) = 2.5 + 3 sin(0.5 * t) and j 3 (t) = 1.2 + 0.3 cos(10t). Take θ = 3. Thus the "new" parameter b = 0.5 cos(0.5t) + 3.5 and b [3,4]. So the resulting system (8) is still chaotic, as depicted in Figure 9. The simulation results of the proposed chaotic secure communication scheme with additive noise are displayed in Figure 10. The synchronization errors e 1 , e 2 and e 3 are shown in Figure  10, from which we can see that the required synchronization has been achieved quickly with our designed controllers (14) and the parameter update rule (15). The information signal f (t) and the recovered onef (t) are displayed in Figure 10. Figure 10 plots the error signal between the original information signal and the recovered one. From Figure 10, it is easy to see that the information signal f (t) is recovered accurately after a very short transient.

Conclusions
In this paper, a novel chaotic secure communication scheme based on GFPS and parameter modulation is proposed. In the transmitter end, the original information signal is modulated into the parameter of the chaotic system and the resulting system still chaotic. There is no constraint posed on the original information signal. In the receiver, we assume that the parameter of the receiver system is uncertain. On the basis of the Lyapunov stability theory, the controllers and corresponding parameter update rule are devised to make the states of two identical Liu chaotic systems with unknown parameter synchronized. Simultaneously the uncertain parameter of the receiver system is also identified. Furthermore, the information signal can also be recovered accurately and fast by applying our secure communication method when additive noise exists in the transmission channel. Numerical simulations show the effectiveness and feasibility of the proposed chaotic secure communication scheme based on GFPS and parameter modulation.