A New 5-D Multistable Hyperchaotic System With Three Positive Lyapunov Exponents: Bifurcation Analysis, Circuit Design, FPGA Realization and Image Encryption

In this work, we describe the model of a new 5-D hyperchaotic system with three positive Lyapunov exponents. Since the maximum positive Lyapunov exponent of the proposed hyperchaotic system is larger than twelve, the new hyperchaotic system is highly hyperchaotic. We also show that the new 5-D hyperchaotic system exhibits multistability with coexisting attractors. Using Multisim, we design an electronic circuit for the new 5-D hyperchaotic system. The hardware implementation of the new 5-D hyperchaotic system is done by applying two numerical methods. From the experimental results of the FPGA-based implementation, we show that the attractors observed in a Lecroy oscilloscope are in good agreement with numerical simulations. To prove the reliability of the proposed system for cybersecurity purposes, we present a new image cryptosystem using our hyperchaotic system. Experimental outcomes show the efficiency and the reliability of our cryptosystem based on the proposed hyperchaotic system.

encryption method based on a 6-D hyperchaotic Lorenz system and performed a circuit simulation to verify the security of the communication scheme. Wang et al. [3] designed a 6-D hyperchaotic system and applied it to complete the signal encryption and decryption circuit design of a secure communication scheme. Peng et al. [4] proposed a new 5-D hyperchaotic system for secure communication based on Micro Controller Unit (MCU). Xiu et al. [5] presented a memristive cellular neural network (CNN) hyperchaotic system and described its hardware circuit design. Li et al. [6] presented a new 5-D memristive hyperchaotic system based on hyperchaotic Lü system and described its extreme multistability. Alibraheemi et al. [7] described the design and simulation of a 4-D hyperchaotic communication system based on the Lorenz chaotic attractor.
In this work, we describe the model of a new 5-D hyperchaotic system with three equilibrium points and deduce that the new hyperchaotic system has three positive Lyapunov exponents (LE). We show that the maximum positive Lyapunov exponent of the proposed hyperchaotic system is larger than twelve. Thus, we observe that the new hyperchaotic system is highly hyperchaotic.
We present a detailed dynamic analysis for the new 5-D hyperchaotic system with bifurcation diagrams and Lyapunov exponents. Bifurcation analysis is useful for understanding the dynamic behavior of hyperchaotic systems with respect to changes in values of the system parameters [8], [9], [10]. We also note that the new hyperchaotic system exhibits multistability with coexisting attractors.
The application of electronic circuits in the hyperchaotic system has been widely explored by researchers such as secure communication system, high frequency generator, robotic, structure optimization of antenna arrays, satellite communications, autonomous unmanned aerial vehicles and sound encryption [11], [12], [13]. Using Multisim, we design an electronic circuit for the implementation of the new 5-D hyperchaotic system and present the circuit simulation results. Circuit design for chaotic and hyperchaotic systems has important applications in engineering [11], [12], [13], [14].
Autonomous and nonautonomous chaotic systems have been implemented with different analog and digital electronic devices. For instance, the recent paper in [15] shows the electronic implementation of an autonomous chaotic and hyperchaotic system made of a Colpitts-Josephson junction like circuit and three physical energy storage tanks, namely CLC. The circuit is simulated using the simulation program with integrated circuit emphasis (SPICE), and also, the authors implement the chaotic system using Arduino UNO board that is based on a microcontroller. In a similar way, the authors in [16], introduce a new 4D hyperchaotic system showing its analog and digital Implementation. The analog implementation is performed using amplifiers, and the digital implementation using a Digital Signal Processor (DSP), which showed good agreement with simulation results. The design of chaotic systems can also be performed using integrated circuit technology [17], however, due to the design challenges when using nanometer technologies, authors prefer reconfigurable hardware. For instance, one decade ago the authors in [18] demonstrated the usefulness of the fieldprogrammable analog array (FPAA) to design chaotic systems, emphasizing that this device provides flexible design possibilities such as reducing the complexity of design, realtime modification, software control and adjustment within the system. Other works that use FPAAs to implement chaotic systems can be found in [19], [20], [21], [22].
Although amplifiers, FPAAs and microcontrollers have demonstrated their usefulness in implementing chaotic systems, the field-programmable gate array (FPGA) device is more appropriate for fast prototyping and verification of chaotic attractors. The FPGA is a programmable device that can be configured to implement the functionality comparable to custom hardware without requiring additional fabrication steps, printed circuit board fabrication, etc. In addition, it facilitates parallel computing because each different computing task can be assigned to a dedicated group of logic blocks. In Sect. VII we show details on the FPGA design of this new 5-D chaotic system with three positive Lyapunov exponents.
Data security acts a vital role in present days, in which digital images represent the familiar model of representing data [23]. The information existing in digital images can be protected via applying one of image encryption and/or data hiding algorithms [24]. Chaotic maps are considered the backbone of designing modern image encryption techniques [12], [25], [26]. In this paper, we presented a new color image cryptosystem based on the presented 5-D hyperchaotic map. Experimental outcomes demonstrate the efficiency and the reliability of our cryptosystem, accordingly, our hyperchaotic system can be dependable in various security applications.

II. A NEW 5-D HYPERCHAOTIC SYSTEM A. MATHEMATICAL MODEL OF THE NEW SYSTEM
In this paper, we propose a new 5-D hyperchaotic system modelled by the dynamics where z 1 , z 2 , z 3 , z 4 , z 5 are the states and a, b, c, d are parameters.
In this work, we establish that the system (1) is hyperchaotic with three positive Lyapunov exponents if we take the initial state as Z (0) = (1, 1, 1, 1, 1) and parameters as (2) 90112 VOLUME 10, 2022 The system (1) stays invariant under the change of coordinates for all values of the system parameters. This shows that the system (1) has rotational symmetry about the z 3 -axis.

B. DYNAMIC PROPERTIES OF THE NEW SYSTEM
Using Matlab, we calculated the Lyapunov exponents of the 5-D system (1) for the initial state Z (0) = (1, 1, 1, 1, 1), (a, b, c, d) = (40,90,16,15) and T = 1E5 seconds as The Kaplan-Yorke dimension of the 5-D system (1) is calculated as follows: From (4), we deduce that the 5-D system (1) is a dissipative hyperchaotic system with three positive Lyapunov exponents. Since the value of maximum Lyapunov exponent (MLE) in Eq. (4) is large, we conclude that the 5-D system is highly hyperchaotic and this property is very useful for applications of the hyperchaotic system in cryptosystems and secure communications.  The equilibrium points of the 5-D system (1) are found by solving the following equations.
Clearly, E 0 = 0 ∈ R 5 is an equilibrium point. A simple calculation shows that there are two more equilibrium points given by Suppose we take the parameters as in the hyperchaotic case (2), i.e. a = 40, b = 90, c = 16 and d = 15.
For this case, λ = a+b a = 3.25. Also, the three equilibrium points of the system (1) are obtained as From the viewpoint of Lyapunov exponents, an n-dimensional hyperchaotic system is more complex when it has more number of positive Lyapunov exponents (n − 2 positive Lyapunov exponents), which indicates that the dynamic of the system can expand in several directions in the phase space. In addition, a large positive Lyapunov exponent means that the system dynamic has a very fast exponential expansion in the phase space, providing stronger hyperchaotic behavior [27]. Moreover, the Kaplan-York dimension measures the complexity of the attractor. A high value of the Kaplan-Yorke dimension corresponds to a highly complex attractor. Based on the above discussions, a strong 5-D hyperchaotic system must have the following three properties: (i) Three positive Lyapunov exponents to ensure that the dynamics of the system expand in three different directions of phase space. (ii) A large positive Lyapunov exponent to ensure a fast exponential expansion in phase space. VOLUME 10, 2022 (iii) A high value of Kaplan-York dimension to ensure a high complexity of the attractor. Table 1 shows the Lyapunov exponents and the Kaplan-Yorke dimension of the new system (1) and those of ten recently reported 5-D hyperchaotic systems. It can be seen that the new system (1) has three positive Lyapunov exponents, contrary to many other reported 5-D systems that have two positive Lyapunov exponents. Also, system (1) has the largest first Lyapunov exponent, which means that its trajectories expand more quickly than those of other reported 5-D systems. Finally, we can see that the new system (1) has a much larger Kaplan-Yorke dimension than that of the most recently reported 5-D hyperchaotic systems. As a result, the new 5-D hyperchaotic system is much stronger than many reported 5-D hyperchaotic systems. It is a big addition to the hyperchaotic systems family. Its properties make it very desirable to use in chaos-based applications.

III. BIFURCATION ANALYSIS OF THE NEW 5-D HYPERCHAOTIC SYSTEM
In this section, the dynamical properties of the 5-D hyperchaotic system (1) are investigated using Lyapunov exponents spectrums, bifurcation diagrams and phases plots. Influence of the system parameters on the behavior of the new highlycomplex hyperchaotic system (1) will be presented when Z (0) = (1, 1, 1, 1, 1).

A. DYNAMICAL BEHAVIOR EVOLUTION WHEN a CHANGES
Dynamical properties of the 5-D system (1) are presented here when the value of parameter a increases in the region [10,40], while the other parameters are fixed as b = 90, c = 16 and d = 12.
Simulation results shows that system (1) can present different behaviors for different values of parameter a.
Bifurcation diagram presented in Figure 3 (a) and Lyapunov exponents spectrum shown in Figure 3 (b) demonstrate that, based on the values of the control parameter a, the 5-D system (1) can exhibit periodic behavior, chaotic behavior with small value of maximal Lyapunov exponent (MLE), hyperchaotic behavior with two positive Lyapunov exponents and high value of MLE or hyperchaotic behavior with three positive Lyapunov exponents and a very high value of MLE.
Define A = [0, 18.5] ∪ [19.2, 20]. When a ∈ A, the MLE of the new 5-D system (1) is zero, implying that system (1) displays periodic behavior with no complexity. We chose the control parameter a = 12. Then, the periodic attractor is plotted in (z 1 , z 2 ) plane in Figure 4 (a). The corresponding Lyapunov exponents are: When a ∈ B, the system dynamics (1) changes from periodic to chaotic via the famous period-doubling route to chaos as shown in Figure 3 (c). The MLE is positive and the Kaplan-Yorke dimension has a fractional value of 2.044, which demonstrates that the system (1) generates complex chaotic behavior for the mentioned region of control parameter. When a = 19, the chaotic attractor is plotted in (z 1 , z 2 ) plane in Figure 4 (b). The corresponding Lyapunov exponents are: When a ∈ [20.4, 32], the 5-D system dynamics (1) changes from chaotic to hyperchaotic with two positive Lyapunov exponents and a high value of MLE. Thus, in this region of control parameters, the system (1) displays a very complex hyperchaotic behavior characterized with the following fractional value of Kaplan-Yorke dimension D KY = 4.219. When a = 25, the chaotic attractor is plotted in (z 1 , z 2 ) plane in Figure 4 (c). The corresponding Lyapunov exponents are: When a ∈ [32,40], the 5-D system dynamics (1) displays hyperchaotic behavior with three positive Lyapunov exponents and a very high value of MLE (about 12) providing extreme complexity of the dynamics.
When a = 39.8, the hyperchaotic attractor of the new 5-D system is plotted in (z 1 , z 2 ) plane in Figure 4 (d) and characterized with the following fractional value of Kaplan-Yorke dimension D KY = 4.178. The corresponding Lyapunov exponents are: Behaviors, Lyapunov exponents values, Kaplan-Yorke dimensions and attractors of the new 5-D system (1) for different values of parameter a are summarized in Table 2.  Simulation results shows that system (1) can present different behaviors for different values of parameter b.
Bifurcation diagram presented in Figure 5 (a) and Lyapunov exponents spectrum shown in Figure 5 (b) demonstrate that, based on the values of the control parameter b, the 5-D system (1) can exhibit periodic behavior, hyperchaotic VOLUME 10, 2022 behavior with two positive Lyapunov exponents, hyperchaotic behavior with three positive Lyapunov exponents or converges to equilibrium point.
When b ∈ [0, 9], from Figure 5 (b), we can see that the MLE of the new 5-D system is zero, implying that the system (1) displays periodic behavior, which is confirmed by bifurcation diagram in Figure 5 (a). When we choose the control parameter b = 4, the periodic attractor of the system (1) is plotted in (z 1 , z 3 )-plane in Figure 6 (a). The corresponding Lyapunov exponents are: When b ∈ [9, 14.5], from Figure 5 (b), we can see that the MLE of the new 5-D system is negative, implying that system (1) converges to equilibrium. Thus, in this region of control parameters, the system (1) displays a stable dynamics. When b = 10, the attractor of the new 5-D system (1) is plotted in (z 1 , z 3 )-plane in Figure 6 (b). The corresponding Lyapunov exponents are: When b ∈ C, the system dynamics (1) changes to hyperchaotic behavior with two positive Lyapunov exponents as shown in Figure 4 (b). Hence, in this region of control parameters, the system (1) displays a complex behavior characterized with the following fractional value of Kaplan-Yorke dimension D KY = 3.282. When b = 16, the hyperchaotic attractor of the new 5-D system (1) is plotted in (z 1 , z 3 )-plane in Figure 6 (c). The corresponding Lyapunov exponents are:   Table 2. Simulation results shows that system (1) can present different behaviors for different values of parameter c.
Bifurcation diagram presented in Figure 7 (a) and Lyapunov exponents spectrum shown in Figure 7 (b) demonstrate that, based on the control parameter c, the 5-D system (1) can exhibit chaotic behavior, hyperchaotic behavior with two positive Lyapunov exponents or hyperchaotic behavior with three positive Lyapunov exponents.
When c ∈ [0, 1.4], from Figure 7 (b) we can see that there is one positive Lyapunov exponent, implying that the system (1) displays chaotic behavior, which is confirmed by bifurcation diagram in Figure 7    system (1) is plotted in (z 1 , z 4 ) plane in Figure 8 (a). The corresponding Lyapunov exponents are: When c ∈ [1.4, 6], the system (1) dynamics change from chaos to hyperchaos with two positive Lyapunov exponents as shown in Figure 7 (b). Thus, in this region of control parameters, the system (1) dynamics expands in two different directions of phase space.
When c = 2, the hyperchaotic attractor of the new 5-D system (1) is plotted in (z 1 , z 4 )-plane in Figure 8 The Kaplan-Yorke dimension is fractional with a value of D KY = 4.189 and the corresponding Lyapunov exponents are: When c ∈ [6, 20], the system (1) exhibits hyperchaotic behaviour with three positive Lyapunov exponents. Hence, in this region of control parameters, the system (1) displays a complex behavior which expands in three different directions of phase space and characterized with the following fractional value of Kaplan-Yorke dimension D KY = 4.186. When c = 20, the hyperchaotic attractor of the new 5-D system (1) is plotted in (z 1 , z 4 ) plane in Figure 8 (c). The corresponding Lyapunov exponents are: Behaviors, Lyapunov exponents values, Kaplan-Yorke dimensions and attractors of the new 5-D system (1) for different values of parameter c are summarized in Table 2. Simulation results show that system (1) can present different behaviors for different values of parameter d.
Bifurcation diagram presented in Figure 9 (a) and Lyapunov exponents spectrum shown in Figure 9 (b) demonstrate that, based on the control parameter d, the 5-D system (1) can exhibit chaotic behavior, hyperchaotic behavior with two positive Lyapunov exponents or hyperchaotic behavior with three positive Lyapunov exponents.
When d ∈ [0, 1.1], the MLE of the new 5-D system is zero, implying that system (1) displays periodic behavior with no complexity. When we chose the control parameter as d = 0.7, then the periodic attractor of the system (1) in (z 1 , z 5 ) plane is plotted in Figure 10 (a). The corresponding VOLUME 10, 2022  Lyapunov exponents are: When d ∈ [1.1.7], system (1) dynamics change from periodic to hyperchaotic with two positive Lyapunov exponents. The MLE is over 4 and the Kaplan-Yorke dimension has a fractional value of 4.100, which demonstrates that system (1) generates complex hyperchaotic dynamics which expands in two different directions of phase space for the mentioned region of control parameter.
When d = 4, the hyperchaotic attractor is plotted in (z 1 , z 5 ) plane in Figure 10 (b), which displays the hyperchaotic behavior of system (1). The corresponding Lyapunov exponents are: When d ∈ [7,15], the system (1) displays hyperchaotic behavior with three positive Lyapunov exponents and a very high value of MLE (over 12) providing extreme complexity of the dynamics.
When d = 14, the hyperchaotic attractor of the new 5-D system (1) is plotted in Figure 10 (c), and characterized with the following fractional value of Kaplan-Yorke dimension D KY = 4.187. The corresponding Lyapunov exponents are:  Table 2.

IV. MULTISTABILITY AND COEXISTENCE OF ATTRACTORS
Multistability or coexisting attractors is a nonlinear phenomenon that means two or more different attractors generate simultaneously from different initial points [38], [39], [40]. By using numerical simulations, we found that the new 5-D system (1) is rich in coexisting attractors. The 5-D system (1) displays coexistence of two periodic attractors, two chaotic attractors or two hyperchaotic attractors for different choices of parameter values.
Since the 5-D system (1) is invariant under the coordinates transformation the appearance of coexisting attractors may occur if we make the appropriate selection of initial points.
Fix the parameters as a = 40, b = 8, c = 16 and d = 12. Then the 5-D system (1) displays two coexisting periodic attractors as shown in Figure 11 (a), where the blue attractor starts from Z 0 and the red attractor starts from W 0 .  Fix the parameters as a = 20.3, b = 90, c = 16 and d = 12. Then the 5-D system (1) displays two coexisting chaotic attractors as shown in Figure 11 (b) starting from the initial points Z 0 (blue) and W 0 (red), respectively. The two coexisting chaotic attractors are characterized by the following values of Lyapunov exponents:  hyperchaotic attractors as shown in Figure 11 (c) starting from the initial points Z 0 (blue) and W 0 (red), respectively. The two coexisting hyperchaotic attractors are characterized VOLUME 10, 2022 by the following values of Lyapunov exponents: Fix the parameters as a = 40, b = 41, c = 16 and d = 12. Then the 5-D system (1) displays two coexisting hyperchaotic attractors as shown in Figure 11 (d) starting from the initial points Z 0 (blue) and W 0 (red), respectively. The two coexisting hyperchaotic attractors are characterized by the following values of Lyapunov exponents:

V. OFFSET BOOSTING CONTROL
Offset boosting control means that the amplitude of the system can be flexibly controlled after introducing a control parameter in one of the states. The offset boosting does not change the qualitative behavior of the system dynamics but rather moves the attractor from its position to another one in a positive direction or negative direction according to the value of control parameter chosen. Since the fifth state variable of the 5-D hyperchaotic system (1), z 5 , only appears in the fourth equation, the new 5-D system is a variableboosting hyperchaotic system, where z 5 can be controllable. Hence,z 5 can be boosted via substituting z 5 by z 5 + p.
For offset-boosting control, the 5-D hyperchaotic system (1) can be rewritten as: where p is an offset-boosting controller.
Various phase portraits of the hyperchaotic attractors boosted on different values of p in the (z 2 , z 5 ) plane are plotted in Figure 12 (a). The hyperchaotic attractor of the 5-D system (27) is boosted in the positive direction for a negative value of p, while the attractor is boosted in a negative direction for a positive value of p. Consequently, the hyperchaotic  signal z 5 can be boosted from a bipolar hyperchaotic signal to a unipolar hyperchaotic signal as shown in Figure 12 (b). This special feature of offset boosting has a great application in secure communication and other fields of engineering.

VI. ELECTRONIC CIRCUIT IMPLEMENTATION OF THE NEW 5-D HYPERCHAOTIC SYSTEM
The analog circuit of the proposed 5-D hyperchaotic system (1) is implemented by adopting capacitors, resistors, operational amplifiers TL082CD and analog multipliers AD633JN. The circuit design of the 5-D hyperchaotic system is given in Figure 1. Electronic circuit implementation of the hyperchaotic system is designed in MultiSim (Version 13.0) for the parameters a = 40, b = 90, c = 16 and d = 15 with the initial condition Z (0) = (1, 1, 1, 1, 1).  For circuit implementation, we rescale the state variables of the hyperchaotic system (1) as follows: The rescaled 5-D hyperchaotic system is given below: VOLUME 10, 2022 Applying the Kirchhoff laws, the circuit presented in Figure 13 is described by the following: Here, Z 1 , Z 2 , Z 3 , Z 4 and Z 5 correspond to the voltages on the integrators U 1A, U 2A, U 4A, U 5A, U 8A, respectively. The values of resistors are considered as The capacitors are selected as MultiSim outputs of the 5-D hyperchaotic system (refseccircuit-trans-eqn2) are presented in Figure 14, which agree with the MATLAB outputs of the new 5-D hyperchaotic system (1) shown in Figure 1.

VII. FPGA IMPLEMENTATION OF THE 5-D HYPERCHAOTIC SYSTEM
The FPGA is a hardware circuit allowing programmability to carry out both logical operations and arithmetic operations. Among its advantages is the great flexibility for fast prototyping and low development cost applications. In the case of chaotic systems, the FPGA implementation requires determining a numerical method and the step-size h to reach the maximum operating frequency. In this manner, the new 5-D chaotic system with three positive LEs, can be solved by applying one-step numerical methods with an appropriate h, as done in [41], where it is mentioned that using a high value of h helps to increase the operating frequency of chaotic/hyperchaotic systems. After selecting a numerical method, one can emulate the chaotic system in an FPGA as already done in [42], [43], [44], [45], [46], [47]. If the chaotic system has different parameter values to be tested, the FPGA is appropriate to reconfigure or reprogram the values. The main difference in FPGA implementations of chaotic systems is mainly measured by the use of hardware resources, which are often divided into configurable logic cells, random access memory (RAM) block, DSP slices, and transceivers. Some multipliers are embedded into the FPGA but in some cases they can be designed using shift registers and adders, as shown in [48], where single-constant-multiplier (SCM) blocks are implemented to generate multi-scroll chaotic attractors. In that work, the authors also show the application of Forward Euler (FE) and Runge Kutta (RK) methods for the FPGA implementation of a chaotic system based on saturated nonlinear function series. In this work, we show the application of FE and Trapezoidal rule. The former is an explicit numerical method and the second an implicit one, which requires the use of an implicit method, as shown below.
The FE method is based on (38), which current iteration index n+1 depends on one-step at iteration n, and is evaluated by f (x n , t n ). Applying this numerical method, the discretized equations of the new 5-D hyperchaotic system are given by (39), from which one can identify arithmetic operations that can be implemented by using multipliers, adders and subtractors.
The second numerical method applied herein is the Trapezoidal rule whose iterative equation is given in (40). As this method requires of evaluating the function at the same current index, i.e. f (x n+1 , t n+1 ), we apply FE to evaluate this value. In this manner, the discretized equations are given in (41), where one can see that more arithmetic operations are needed, so that it requires more resources in the FPGA implementation.
The synthesis of all the blocks to perform the arithmetic operations can be designed by adopting a computer arithmetic based on fixed-point format. In this case, each real value is represented with the format 16.48 (64 bits), using one bit for the sign, 15 bit to represent the integer part, and 64 for the fractional part. It is worthy mentioning that this 16.48 format is established after observing the amplitudes of the state variables. For example, simulation results for this new 5-D hyperchaotic system taking into account all multiplications among state variables, provide the amplitude range approximately between [−24000,23500], so that 15 bits are enough to represent up to 32768 in the integer part, and one bit is used for the sign, given 16 bits in the format 16.48. The high level description of the main blocks for the implementation of the new 5-D hyperchaotic system, is shown in Fig. 17. As one sees, the values of the coefficients a, b, c, d are stored in a read only memory (ROM), which can be reprogrammed if they are optimized. The iterations are controlled by a state machine to process the values are iteration n to give the values at current iteration n + 1. Those results at n + 1 are always available at the registers and feedback through the multiplexer blocks, which also control the initial conditions. The implementation applying the Trapezoidal method is performed in a similar manner, but it requires of an explicit method, as already shown in [48] applying RK method and FE as predictor. In this manner, the hardware resources are listed in Table 3. One can appreciate that the Trapezoidal method doubles the logic elements, as one can infer observing (40) and (41). The requirement of multipliers also increase in the Trapezoidal method. Finally, as the state machine is designed to control the same clock cycles, the latency is the same.

VIII. APPLICATION TO IMAGE CRYPTOSYSTEM
Data security acts a vital role in present days, in which digital images represent the familiar model of representing the data.  Chaotic maps are considered the backbone of designing modern image encryption mechanism. In this section, we adapt the hyperchaotic system to be iterative and applicable in  modern security proposes as expressed in Eq. (42).  FIGURE 19. Dataset of images and its resulted cipher images using our cryptosystem.

A. ENCRYPTION PROCEDURE
In this section, we present a new color image encryption mechanism using the presented 5-D hyperchaotic map (42). In the proposed algorithm, the hyperchaotic system is employed to generate five chaotic sequences (Z1, Z2, Z3, Z4, and Z5), in which the first two chaotic sequences (Z1 and Z2), are utilized to construct two permutation boxes for permutating the plain image (PG), then the other three sequences (Z3, Z4, and Z5) are operated to substitute the permutated image (PrG). To ensure the plain image sensitivity of our encryption technique, SHA256 hashing algorithm is applied on the substituted image (SG), then utilizes the hash value for updating the initial values (z(1) 0 , z(2) 0 , z(3) 0 , z(4) 0 , and z(5) 0 ) of the presented hyperchaotic map. Note that the generated hash value does not depend only on the plain image (PG) but also on the primary conditions and control parameters (z(1) 0 , z(2) 0 , z(3) 0 , z(4) 0 , z(5) 0 , a, b, c, and d) of the hyperchaotic system. The updated initial values (z(1) u , z(2) u , z(3) u , z(4) u , and z(5) u ) are utilized to iterate the hyperchaotic system for generating another five chaotic sequences, in which the first two chaotic sequences are utilized to construct a substitution box (S-box), then the other three sequences and the generated S-box are utilized to construct the final cipher image. For more illustration, the sketch of the encryption algorithm is outlined in Figure 18 and the details of the encryption technique are given in Algorithm 1.

B. SECURITY ANALYSIS
The quality of a cryptosystem to encrypt a document depends essentially on its performance and resist security attacks such as differential cryptanalysis, statistical cryptanalysis, brute force attacks, etc. These fundamental effects are investigated in this section.
To assess the proposed encryption approach, a personal computer with 4GB of RAM and Intel Core TM 2 Duo CPU 3GHz and equipped with MATLAB R2016b. The experimental dataset of images is taken from SIPI database [49] and labeled as Sailboat, Mandrill, Airplane, and House ( Figure 19), in which the dimensions of each color image is 512 × 512. The initial key utilized for iterating the hyperchaotic system are set as z (1)

1) ANALYSIS OF CORRELATION COEFFICIENTS
To evaluate the meaningful of cipher images, the test of correlation coefficients is utilized, which can be stated mathematically as follows: where c j , p j are refer to the values of adjoining pixels and T refers to the total number of adjoining pixel pairs in 90126 VOLUME 10, 2022   each direction. Commonly, the values of CC for plain images are very near to one, while its values for cipher images with well-designed image cryptosystem are very near to zero. To calculate CC values for our cryptosystem, we selected randomly 10,000 pairs of neighboring pixels in each direction. Table 4 provides the outcomes for CC, in which CC values    as follows: where CG1, CG2 are two cipher images for one original image with modification in 1-bit, and T refers to the complete number of pixels. The outcomes of UACI and NPCR testes are displayed in Table 5 and confirm that the presented cryptosystem owns high sensitivity for slight pixel alterations in the plain image.

3) HISTOGRAM ANALYSIS
Histogram test is applied to evaluate the frequency of pixel values in an image, where histograms for different plain images are distinct from each other's, while histograms for cipher images generated from a well-designed cryptosystem ought to be similar with each other for resisting statistical attacks. Figure 23 parades the histograms of Sailboat image before and after ciphering, where the histograms for the plain image are dissimilar with each other whereas the plots of Enc-Sailboat image are similar with each other. However, we lack a quantifiable measure to identify the similarity of histograms such as the Chi-square test (χ 2 ) which can be described mathematically as follows: here s points to the image size and r j states the frequency of pixel value j. By assuming the significant level is α = 0.05, then χ 2 α (255) = 293.32. For an image, when the χ 2 value is fewer than χ 2 α (255), this refers that the given image holds a similar histogram distribution; else, the image owns a dissimilar distribution. The outcomes of Chi-square test are displayed in Table 6, in which all χ 2 values for all ciphered images is fewer than χ 2 α (255). Thus, the presented cryptosystem has the ability to resist histogram analysis attacks.

4) ENTROPY ANALYSIS
To assess the randomness of an image, information entropy test is applied, which can be given as follows: here p(x k ) points to the probability of x k . Any grey scale image has 2 8 probable values, accordingly the ideal value of entropy is equal to 8. Hence, the entropy value of the cipher images generated using a well-designed technique must be very near to the optimal value. The outcomes of the entropy test are displayed in Table 7, where all values for cipher images are very near to the ideal value. Therefore, the suggested cryptosystem has the ability to withstand statistical analysis originated from entropy attacks.

5) DATA LOSS ANALYSIS
Most telecommunication channels suffer from data loss attacks. Therefore, well-designed cryptosystems should have the ability to retrieve secret data from deficient cipher images.
To assess the proposed cryptosystem contra these attacks, we cut out some slices of a cipher image, then attempt to decipher the deficient encrypted image. Figure 24 displays the outcomes of data loss attacks, where the original image is retrieved efficiently from the deficient encrypted image without losing information in the cut slice.

6) KEY SENSITIVITY ANALYSIS
Any cryptosystem has a secret key used in the ciphering and deciphering procedure. The utilized key must be sensitive to tiny modification in the secret key. To assess the key sensitivity of our cryptosystem, the cipher Enc-Sailboat image was deciphered many times with tiny alternations in the primary key parameters as stated in Figure 25.

C. COMPARATIVE ANALYSIS
To prove that our cryptosystem possesses high efficiency and security compared to most recent encryption algorithms, Table 8 shows a simple comparison in average values of UACI, NPCR, Chi-square, entropy, and correlation. It is obvious that the presented cryptosystem is effective compared to other encryption algorithms.

IX. CONCLUSION
This work described the model of a new 5-D hyperchaotic system with three equilibrium points and deduced that the new hyperchaotic system has three positive Lyapunov exponents. Since the maximum positive Lyapunov exponent of the proposed hyperchaotic system is larger than twelve, we noted that the new hyperchaotic system is highly hyperchaotic. We also showed that the new 5-D hyperchaotic system exhibits multistability with coexisting attractors. Offset boosting control is also derived for the new 5-D hyperchaotic system. Using MultiSim, we designed an electronic circuit for the implementation of the new 5-D hyperchaotic system and present the circuit simulation results. The FPGA implementation of the new 5-D hyperchaotic system has been performed applying an explicit numerical method (FE) and an implicit one (Trapezoidal). We provided the discretized equations applying both methods to observe the arithmetic operations, so that the hardware resources given in Table 3 show that the Trapezoidal method doubles the logic elements, and requires more multipliers. Also, the experimental observation of the attractors show good agreement among FPGA implementation, circuit design and numerical simulations. Finally, the proposed 5-D hyperchaotic system can be utilized in designing reliable cybersecurity applications. As an example of the cryptographic application for the presented 5-D hyperchaotic map, we proposed a new color image cryptosystem, in which its experimental results proved its efficiency. Sultan University, Riyadh, Saudi Arabia. He is the author or coauthor of more than 200 papers, including refereed IEEE/ACM/Springer/Elsevier journals, conference papers, books, and book chapters. He had many books, more than ten books, in several publishers for process in Springer, IET, CRC press, IGI-Global, Wiley, and IEEE. His research interests include multimedia content encryption, secure wireless communication, the IoT, applied cryptanalysis, perceptual cryptography, secret media sharing, information hiding, biometrics, forensic analysis in digital images, and quantum information processing. He is a fellow of the Academy of Scientific Research and Technology, Egypt. He received many awards, including the State Encouragement Award in Engineering Sciences in 2016, Arab Republic of Egypt; the Best Ph.D. Student Award from the Harbin Institute of Technology in 2013; and the Young Scientific Award, Menoufia University, Egypt, in 2014. He is an editor/guest editor in several reputed SCI journals.
BASSEM ABD-EL-ATTY received the B.S. degree in physics and computer science, the M.Sc. degree in computer science, and the Ph.D. degree in computer science from Menoufia University, Egypt, in 2010, 2017, and 2020, respectively. He is currently an Assistant Professor with the Faculty of Computers and Information, Luxor University, Egypt. He is the author or coauthor of more than 30 papers, including refereed articles published by IEEE, Elsevier, and Springer journals, conference papers, and book chapters. His research interests include quantum information processing and image processing. He is a reviewer in a set of reputable journals in Elsevier and Springer.
CIRO FABIAN BERMUDEZ-MARQUEZ received the B.Sc. degree from the Benemérita Universidad Autónoma de Puebla (BUAP), Mexico, in 2020. He is currently pursuing the M.Sc. degree in electronics with the Instituto Nacional de Astrofísica, Óptica y Electr onica (INAOE), Mexico. His research interests include chaotic systems, optimization by metaheuristics, embedded systems, and digital signal processing.
IBRAHIM MOHAMMED SULAIMAN received the Ph.D. degree in fuzzy systems from the Universiti Sultan Zainal Abidin (UniSZA), Malaysia, in 2018. Since 2021, he has been a Lecturer with the Universiti Utara Malaysia, Malaysia. He has published research articles in various international journals and attended international conferences. His research interests include numerical research, fuzzy nonlinear systems, and unconstrained optimization.
ALIYU MUHAMMED AWWAL received the B.Sc. degree from Gombe State University, the M.Sc. degree from Bayero University Kano, and the Ph.D. degree in applied mathematics from the King Mongkut's University of Technology Thonburi (KMUTT). He has authored or coauthored a number of research articles in high impact journals. His research interest includes iterative algorithms for solving nonlinear problems, such as numerical optimization problems, nonlinear least squares problems, and system of nonlinear equations with applications in signal recovery, image deblurring, and motion control.
POOM KUMAM (Member, IEEE) received the Ph.D. degree in mathematics from Naresuan University, Thailand. He is currently a Full Professor with the Department of Mathematics, King Mongkut's University of Technology Thonburi (KMUTT). He is also the Head of the Fixed Point Theory and Applications Research Group, KMUTT, and also with the Theoretical and Computational Science Center (TaCS-Center), KMUTT. He is also the Director of the Computational and Applied Science for Smart Innovation Cluster (CLASSIC Research Cluster), KMUTT. He has provided and developed many mathematical tools in his fields productively over the past years. He has over 800 scientific articles and projects either presented or published. He has editorial board journals more than 50 journals and also he delivers many invited talks on different international conferences every year all around the world. His research interests include fixed point theory, variational analysis, random operator theory, optimization theory, approximation theory, fractional differential equations, differential game, entropy and quantum operators, fuzzy soft set, mathematical modeling for fluid dynamics, inverse problems, dynamic games in economics, traffic network equilibria, bandwidth allocation problem, wireless sensor networks, image restoration, signal and image processing, game theory, and cryptology.