#### Abstract

This paper deals with the projective synchronization (PS) of two identical discrete-time generalized four-dimensional (4D) hyperchaotic Henon maps using a master-slave configuration. A discrete sliding mode controller (DSMC) scheme is proposed to synchronize the master and the slave systems. The performance of the controlled systems is simulated; the simulation results indicate that the proposed controller works well. In addition, a secure communication scheme is proposed based on the developed control scheme. The validity of the proposed scheme is tested by transmitting an image and simulating the results. The simulation results clearly indicate the effectiveness of the proposed secure communication scheme.

#### 1. Introduction

Chaos synchronization refers to the process wherein two or more chaotic systems adjust a given property of their motion to a common behavior by either coupling or forcing [1]. Depending upon the particular coupling configuration, the process leading to synchronized states is divided into two cases. These two cases are unidirectional coupling and bidirectional coupling. For the case of bidirectional coupling, both subsystems are coupled with each other. The coupling factor induces an adjustment of the rhythms onto a common synchronized manifold. As a result, a mutual synchronization behavior will be induced. This situation appears in nonlinear optics such as lasers [2], in physiology [3] or between interacting neurons [4].

On the other hand, the unidirectional coupling case is totally different from the bidirectional situation. In the unidirectional coupling, two subsystems form a global system using a drive-response (master-slave) configuration. The response system evolution will be driven by the drive system. Thus, the response system will be forced to follow the dynamics of the drive system and hence the latter will act as an external chaotic forcing of the response. The master-slave configuration of chaos has many applications in real life such as telecommunications [5], energy resource systems [6], electrical drives [7], gyro systems [8], satellites [9], and chaos authentication [10]. In 1990, ott et al. [11] were able to control chaos by forcing a chaotic system to follow a desired behavior; this control strategy is known as OGY method. In the same year, Peccora and Caroll [12] were able to control chaos through the method of chaos synchronization. After that, various schemes were successfully applied to chaos and hyperchaos synchronization for both continuous-time systems [13–21] and discrete-time systems [22–29].

The most common types of synchronization in interacting chaotic systems include complete synchronization (CS) [1], phase and antiphase synchronization [30], lag synchronization (LS) [31], generalized synchronization (GS) [32], anticipating synchronization (AS) [33], projective synchronization (PS) [34], and Q-S synchronization [24].

Chaotic signals have special properties that satisfy the requirements for secure communication systems, such as irregularity, aperiodicity, broadband, and unpredictability. Because of these properties, several communication schemes using chaos were proposed and extensively studied [35–37].

Several methods were proposed in the literature for designing chaotic communication systems. One proposed method is chaotic masking where the message is added to the states of the transmitter [38, 39]. In this case, synchronization is needed at the receiver end to extract the message by performing subtraction from the synchronized states of the receiver. Another proposed method is chaotic modulation [40]. In this method, the message is used to modulate one parameter of a chaotic system at the transmitter end; then the message is recovered at the receiver end through synchronization. Also, the method of chaotic shift keying was proposed for transmitting digital messages [5, 41]. Synchronization is used in this method to reconstruct the message at the receiver end. Other secure communication methods include the inverse system approach [42] and chaos control method [43]. In all the previously mentioned methods, the transmitter and the receiver are synchronized to be able to reconstruct the message at the receiver end. Hence, synchronization is important to regenerate the chaotic carrier signal and thus recover the transmitted message.

Nowadays using computers and DSP chips to implement controllers of discrete-time systems has become very common and simple. Hence, several researchers have designed and implemented discrete-time secure communication systems [44–46].

The design of chaos based secure communication schemes has been tackled by many researchers. Several communication schemes were proposed in the unidirectional configuration [36] and the bidirectional configuration [47] for continuous-time systems [36, 48] and for discrete-time systems [47], using chaos synchronization [36], hyperchaos synchronization [45], projective synchronization [47], and other types of synchronization [36]. However, several techniques can be combined to further enhance the security and to make it easier to implement the proposed schemes.

This paper presents a discrete time sliding mode control law for the projective synchronization problem of discrete-time hyperchaotic systems. The proposed control law is then used to design a secure communication scheme. Hyperchaotic systems contain more complex dynamical characteristics than chaotic systems since the attractor has more than one positive Lyaponuv exponent [49]. Therefore, using hyperchaos synchronization to design secure communication schemes can enhance security. The proposed sliding mode controller uses one control input and gives a robust control method for hyperchaos synchronization in the master-slave configuration. In addition, when using projective synchronization, the states of the master and the slave systems synchronize up to a scaling factor [34]. This scaling factor can be used as an additional enhancement to the security of the communication scheme. Therefore, a key contribution of this paper the combination of projective synchronization with hyperchaos to enhance the security of the proposed communication scheme. Hence, the proposed secure communication scheme has better security performance and it is easier to implement.

The paper is organized as follows. The 4D hyperchaotic Henon map is described in Section 2. Section 3 presents the synchronization problem of two 4D hyperchaotic Henon maps using a discrete sliding mode controller; simulation results are presented. A secure communication using the synchronization of the 4D hyperchaotic Henon maps is developed in Section 4. Finally, some concluding remarks are given in Section 5.

#### 2. The 4D Hyperchaotic Henon Map

##### 2.1. Description of the 4D Hyperchaotic Henon Map

The discrete-time generalized four-dimensional hyperchaotic Henon map is described by a fourth order system of difference equations such as [50] where , , , and are the state variables of the system and are the parameters of the system. The system given by (1) exhibits chaotic behavior for 7 and . The plots of the state trajectories of the 4D hyperchaotic Henon map are shown in Figure 1.

The method described in the work of Geist et al. [51] is used to calculate the Lyapunov exponents for the 4D Henon map. The calculation was performed by running the system given by (1) for 100000 iterations. The Lyapunov exponents were found to be Since there exists three positive Lyapunov exponents, then the 4D Henon map is a hyperchaotic system [52].

This paper deals with the synchronization of two 4D huper-chaotic Henon maps using a master-slave configuration. Therefore, the following sub-section presents the model of the error system.

##### 2.2. Model of the Error System

Let the master system be defined as the 4D Henon hyperchaotic map as follows:where , and are the state variables of the master system and are the parameters of the system.

The slave system is defined as the 4D Henon hyperchaotic map as follows:where , , , and are the state variables of the slave system. Note that a controller for the system which is represented by is added to the first difference equation of the slave system. This controller will be designed for the purpose of the projective synchronization of the master and slave systems; i.e., , where is a design parameter.

Define the vectors and such that andAlso, define the error as follows: whereUsing (3) and (4), the error dynamics can be written as follows:The objective of this paper is to synchronize the master and the slave system by forcing the state vector of the slave system (4) to track the scaled state vector of the master system (3); i.e., . This is done by forcing the error to converge to as . A sliding mode controller will be used for this purpose. The design of this controller will be discussed in the next section.

#### 3. Synchronization of Two 4D Hyper-Chaotic Henon Maps Using a Sliding Mode Controller

##### 3.1. Design of the Controller

The first step in the design of a sliding mode controller involves choosing the sliding surface. Since there is one control input in the slave system, then we need to choose one sliding surface.

Let , , and be design parameters such that the roots of the polynomial are located inside the unit circle. The sliding surface is chosen such thatAlso, define the sign function such thatThe subsequent development requires that we define the concepts of quasi-sliding mode (QSM) and the quasi-sliding mode band (QSMB). These concepts were defined in [53]. The definitions are based on the following three attributes that the desired state trajectory of a discrete variable structure (VSC) system should have the following:

*Attribute 1 (A1)*: starting from any initial state, the trajectory will move monotonically toward the switching plane and cross it in finite time.

*Attribute 2 (A2)*: once the trajectory has crossed the switching plane for the first time, it will cross the plane in every successive sampling period, resulting in a zigzag motion about the switching plane.

*Attribute 3 (A3)*: the size of each successive zigzagging step is nonincreasing and the trajectory stays within a specified band.

*Definition 1 (see [53]). *The motion of a discrete VSC system satisfying attributes A2 and A3 is called quasi-sliding mode (QSM). The specified band width contains the QSM is called the quasi-sliding mode band (QSMB) and it is as follows: where is the width of the band.

Let the control parameters be such that , , , , , , and .

The following theorem gives the main result of the paper.

Theorem 2. *The sliding mode controller,when applied to error system (9) guarantees the convergence of the errors to zero as .*

*Proof. *Using the sliding surface given by (10) and using the dynamic model of the errors (9) and the control law given by (13), we obtainEquation (14) corresponds to the double power reaching law defined in [54]. Note that (14) satisfies the reaching and existence condition given by which can be decomposed into the following two inequalities:The above two inequalities can be proven as follows:And,Since and , then .

Using inequalities (17) and (18), it follows that (14) satisfies the condition outside the QSMB. Hence, it can be concluded that reaches zero in finite time.

On the sliding surface, , we haveDefine the error such that Equations (9) and (19) allow us to obtain the following reduced order linear system: whereSince the control parameters , , , and are chosen such that the matrix is stable (i.e., the eigenvalues of are inside the unit circle), then the errors , , and converge to zero as . Using (19), it can be concluded that the error will also converge to zero as . Therefore, the errors , , , and converge to zero as .

The convergence of the errors , , , and to zero as guarantees that , , , and converge to , , , and , respectively, as . Hence, the states of the master and the slave hyper-chaotic systems are synchronized.

##### 3.2. Simulation Results

The sliding mode controller given by (13) is applied to the hyperchaotic 4D Henon-maps given by (3) and (4). The MATLAB software is used to simulate the performances of the controlled systems. To obtain the hyperchaotic behavior of the 4D hyperchaotic Henon map, the parameters and are taken to be and , respectively. The initial conditions of the master system (3) are taken to be , , , and . On the other hand, the initial conditions of the slave system (4) are taken to be , , , and . The projective synchronization factor is . The parameters of the controller are chosen such that , , , , , , , , , and . The performance of the system is simulated from till . For , the performance of the system is simulated with . Then, for the controller is applied to the slave system. In addition, to avoid the occurrence of chattering, the saturation function is used to replace the discontinuous sign function in the control law such that , where is a small positive scalar which can be chosen to get a very good approximation.

The simulation results are presented in Figures 2–11. The plots of the states of the hyperchaotic 4D Henon maps versus are depicted in Figures 2–5. The plots of the errors versus are shown in Figures 6–10. The controller versus is shown in Figure 10. Also, the strange attractors of the master and slave systems are shown in Figure 11.

Figures 2–5 clearly show that the two hyperchaotic Henon maps are synchronized after the application of the proposed sliding mode controller. Figures 6–9 indicate that the errors for the hyper-chaotic 4D Henon maps converge to zero as tends to infinity. Hence, the simulation results show that the proposed sliding mode controller works well as it is able to synchronize the master and the slave systems.

To show the effects of using the saturation function instead of the sign function in the controller, we present the error versus when using the sign function (Figure 12) and the error versus when using the saturation function (Figure 13). It can be seen from these two figures that the usage of the saturation function has greatly reduced the chattering.

##### 3.3. Robustness of the Proposed Controller

To study the robustness of the proposed controller, we will assume that there is a mismatch between the parameters and of the 4D hyperchaotic Henon maps and the parameters and of the controller. We will denote the parameters and of the controller as and such that and .

Using the parameters and , the sliding mode controller (13) becomesThe sliding mode controller given by (23) is applied to the chaotic Henon maps given by (3) and (4) for different values of and . The initial conditions and the other parameters of the controller are the same as in the previous subsection.

Figure 14 depicts the errors , , , and versus when using the sliding mode controller and with and (case 1). Figure 15 shows the errors , , , and versus when using the sliding mode controller and with and (case 2).

Therefore, Figures 14-15 indicate that the proposed controller works well when there is a mismatch between the parameters of the controller and those of the 4D hyperchaotic Henon maps. Thus, it can be concluded that the simulation studies indicate that the developed sliding mode control scheme is robust with respect to mismatches between the parameters of the controller and those of the the 4D hyperchaotic Henon maps.

#### 4. A Secure Communication Scheme Using the 4D Hyperchaotic Maps

##### 4.1. Overview of the Secure Communication Scheme

In this section, the previously proposed approach of synchronization is used to implement a secure communication scheme. The chaotic masking method, where the message signal is added to a chaotic carrier, will be used. A block diagram depiction of the communication scheme is presented in Figure 16.

The different steps involved in the chaotic masking method are the following.

At the transmitter side, the master system is represented by a system of difference equations as follows:

The information message to be transmitted, , is added to the states of the master system to obtain the masked signal where is the state vector, is an matrix, and is a scaling vector.

The masked signals are multiplied by the projective synchronization factor and then sent through a public channel.

The public channel will add noise to the transmitted signals and thus the signals are denoted by .

At the receiver side, the slave system is represented by the following system of difference equations: where is the controller which will be used to synchronize the master and slave systems.

The previously proposed controller will then be used to synchronize the master and the slave systems. The error between the master and slave systems is defined as

To recover the information message, the states of the slave system are subtracted from the states of the received signal and then one obtain the information signal corrupted with some additive noise due to the public channel. Hence we get

##### 4.2. A Secure Communication Scheme Based on the 4D Hyper-Chaotic Henon Maps

The communication scheme is used to send confidential information using 4D Henon maps as depicted in Figure 16. In this case, the master system is defined as the 4D hyperchaotic Henon map as follows:The matrix in (25) is chosen to be and the scaling vector in (25) is such that , so that the information is added to the first state of the master system. The combined signal is then sent through a public channel.On the other hand, the slave system is defined as the 4D hyperchaotic Henon map as follows:Define the errors , , , and such thatwhere , , , and are the transmitted signals corrupted with noise.

We will use the proposed sliding mode controller designed previously to synchronize the master and the slave 4D hyperchaotic Henon map.

Let the sliding surface be such that Using the results presented in Section 3, we will have the following proposition.

Proposition 3. *The following sliding mode controller where , , , and are given by (33) and , , , , , , and guarantee that the errors , , , and converge to zero as .*

The sliding mode controller given by (35) is applied to the systems given by (29) and (32). The MATLAB software is used to simulate the performances of the controlled systems. The initial conditions are taken as , , , and , , , . The parameters of the controller are chosen such that , , , , , , , , , and . The scaling factor . The projective synchronization factor is given by . The transmitted information is an RGB (red, green, and blue) image as shown in Figure 17.

The RGB image shown in Figure 17 is represented by an matrix with values M=225, N=225, P=3. This matrix is converted into a one-dimensional array which can be expressed as follows: . The sequence is converted into a discrete signal and then added to the first state of the hyperchaotic master system.

Since the information will be transmitted through a public channel, an additive white Gaussian noise (AWGN) is added to the transmitted signal with a Signal-to-noise ratio (SNR). The simulations are presented in Figures 18–23. Figure 18 depicts the recovered image when using the sliding mode controller. Figure 19 shows the states and versus ; the errors , , , and versus are shown in Figures 18–23, respectively.

Therefore, it is clear that the proposed communication scheme works well. Hence, it can be concluded that the projective synchronization of two 4D Henon maps can be used for secure communication purposes.

#### 5. Conclusion

The projective synchronization problem is investigated for the hyperchaotic 4D Henon maps. A discrete sliding mode control scheme is proposed to achieve projective chaos synchronization by forcing the errors between the states of the master and the slave systems to converge to zero. The simulation results indicate that the proposed controller works well. Then, the proposed control scheme is used to design a secure communication scheme using 4D Henon maps synchronization. The proposed scheme is validated by transmitting an image through a public channel and recovering the sent image. The simulation results show that the proposed secure communication scheme works well.

Future work will address the usage of other nonlinear control techniques to synchronize discrete chaotic and hyperchaotic systems. Also, different methods to transmit information using discrete chaotic and hyperchaotic systems will be analyzed. Moreover, other transmission methods will be investigated to further enhance the security of communication schemes.

#### Data Availability

All the data was presented in the paper.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.