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A Novel 3D Chaotic System With Line Equilibrium: Multistability, Integral Sliding Mode Control, Electronic Circuit, FPGA Implementation and Its Image Encryption

This paper announces a novel three-dimensional chaotic system with line equilibrium and discusses its dynamic properties such as Lyapunov exponents, phase portraits, equilibrium points, bifurcation diagram, multistability and coexisting attractors. New synchronization results based on integral sliding mode control (ISMC) are also derived for the new chaotic system with line equilibrium. In addition, an electronic circuit implementation of the new chaotic system with line equilibrium is reported and a good qualitative agreement is exhibited between the MATLAB simulations of the theoretical model and the MultiSim results. We also display the implementation of the Field-Programmable Gate Array (FPGA) based Pseudo-Random Number Generator (PRNG) by using the new chaotic system. The throughput of the proposed FPGA based new chaotic PRNG is 462.731 Mbps. Randomness analysis of the generated numbers has been performed with respect to the NIST-800-22 tests and they have successfully passed all of the tests. Finally, an image encryption algorithm based on the pixel-level scrambling, bit-level scrambling, and pixel value diffusion is proposed. The experimental results show that the encryption algorithm not only shuffles the pixel positions of the image, but also replaces the pixel values with different values, which can effectively resist various attacks such as brute force attack and differential attack.

The modelling of chaotic systems with infinite number of equilibrium points is an important research topic in the chaos literature ( [17], [18]). Especially, finding of chaotic systems possessing line equilibrium of points has attracted significant attention in the chaos literature ( [19]- [27]). Jafari et al. [19] implemented a systematic search algorithm to find a gallery of chaotic system with line equilibrium. Kingni et al. [21] constructed two family chaotic system with line equilibrium and hyperbola curve equilibrium. Moysis et al. [22] designed chaotic flow with line equilibrium and application to communications using a descriptor observer. Sambas et al. [23] presented a simple chaotic system with line equilibrium and synchronization by passive control method. Nazarimehr et al. [24] proposed multi-character dynamical systemlike chaotic and hyper-chaotic attractors without any equilibrium, with a line of equilibria or with unstable equilibrium. Chen et al. [25] investigated chaotic behavior in a 3D chaotic system with line equilibrium based on Adomian decomposition. Bao et al. [28] reported a 5-D two-memristor-based chaotic system with plane equilibrium.
Sliding Mode Control (SMC) is attractive for nonlinear systems due to its robustness for both parametric and nonparametric uncertainties. SMC is a well-known method for the synchronization of chaotic systems. However, the invariance of SMC is not guaranteed in a reaching phase. Integral Sliding Mode Control (ISMC) eliminates the reaching phase such that the invariance is achieved in an entire system response [29]. Hence, ISMC is also a popular method for the control and synchronization of chaotic systems. The authors in [29] propose the combination of composite nonlinear feedback (CNF) and ISMC methods for chaos synchronization of a class of uncertain chaotic systems with multiple time-varying delays, Lipschitz nonlinearities and parametric disturbances. In [30], the fixed time integral sliding mode controller with the integral expression, and utilize the continuity of fixed time expression is proposed for suppression of chaotic power systems. The integral sliding mode control (ISMC) scheme is presented in [31] for Hindmarsh-Rose neuron model and two-dimensional fractional-order chaotic FitzHugh-Nagumo neuron model. In [32], the integral sliding mode control theory based on numerically solving a state-dependent Riccati equation is used to design nonlinear feedback control for wind energy conversion systems with permanent magnet synchronous generators.
Most of the chaos-based image encryption schemes are generally composed of two main stages: permutation and diffusion. Numerous chaos-based image encryption algorithms have been studied in many works in the chaos literature ( [33]- [37]). Ping et al. [33] proposed a new block image encryption algorithm. It uses a digit-level permutation and block diffusion with the Henon map. Wu et al. [34] presented the effective image encryption scheme integrating a 4D hyperchaotic system, pixel-level filtering with variable kernels, and DNA-level diffusion. They showed that the PFDD has reliable security keys and is capable of resisting types of attacks [34]. Chen et al. [35] designed image encryption using high-speed scrambling and pixel adaptive diffusion for medical image encryption. It was shown that the proposed improved encryption algorithm not only maintained the merit of the original work but also effectively resisted the chosen plain image attack [35]. Li et al. [36] studied image encryption scheme based on the architecture of bit-level scrambling and multiplication diffusion. It was shown that the proposed encryption scheme significantly improves the security by disturbing known-plain text and chosen-plain text attacks [36]. Zhou and Wang [37] constructed a 5D conservative hyperchaotic system and closed-loop diffusion between blocks for image encryption analysis.
In this paper, a new 3D chaotic system with line equilibrium is reported. The new chaotic system has three quadratic nonlinear terms and an absolute function nonlinearity. We show that the equilibrium set of the new 3D chaotic system is the z-axis of R 3 . Since the new chaotic system has an infinite number of equilibrium points, it belongs to the class of hidden chaotic attractors ( [38], [39]). We have carried out a detailed bifurcation analysis of the new chaotic system with the help of bifurcation diagrams and Lyapunov exponents. Bifurcation analysis of chaotic systems is very useful to understand their qualitative properties ( [40], [41]). The new chaotic system exhibits multistability and coexisting chaotic attractors. Multistability is an important property of a chaotic system which refers to the coexistence of chaotic attractors for the same values of the system parameters but different values of the initial conditions ( [39], [42], [43]). Zhang et al. [43] reported multi-stability in a modified Duffing-Rayleigh system with a piecewise quadratic function.
As a control application, an integral sliding mode control (ISMC) is designed for the achieving global asymptotic synchronization of the proposed chaotic system with itself as master and slave systems. We implement the new chaotic system by using a field-programmable gate array (FPGA) and detail an application for Pseudo-Random Number Generators (PRNGs) based on the new chaotic system. We also apply the new chaotic system to generate a new image encryption algorithm based on the pixel-level scrambling, bit-level scrambling, and pixel value diffusion. As an engineering application, we build an electronic circuit design of the new chaotic system with line equilibrium using MultiSim (Version 13.0). Circuit design of chaotic systems is very useful for various engineering applications ( [44]- [46]). As another engineering application, we devise a new image encryption algorithm based on the new chaotic system with line equilibrium.
This paper is organised as follows. Section 2 describes the dynamics and basic properties of the new chaotic system. Section 3 details the bifurcation analysis, multistability and coexisting attractors of the new chaotic system. Section 4 presents control results using Integral Sliding Mode Control (ISMC) for the synchronization of the new chaotic system. Section 5 details the FPGA implementation and Pseudo-Random Number Generator (PRNG) based on the new chaotic system. Electronic circuit using MultiSim (Version 13.0) of the new chaotic system is given in Section 6. Section 7 describes the image encryption analysis based on the new chaotic system. Section 8 contains the conclusions of this work.

II. MATHEMATICAL MODEL OF THE NEW CHAOTIC SYSTEM
In this paper, we report a new 3-D chaotic system given by the following dynamics: which has three quadratic nonlinear terms and an absolute function nonlinearity. Moreover, in equaton (1), x, y, z denote the states and a, b are the constant parameters. We shall show that the system (1) is chaotic when the system parameters take the values (a, b) = (2.6, 1.5). ( For numerical simulations, we take the initial conditions as With the parameter values as in equation (2) and the initial conditions as in equation (3), the Lyapunov exponents of the new chaotic system in equation (1) are calculated using Wolf algorithm [47] for T = 1E4 seconds as Since the spectrum in equation (4) of Lyapunov exponents of the new chaotic system (1) has a positive Lyapunov exponent, it follows that the new chaotic system (1) is chaotic. Also, the sum of the Lyapunov exponents of the system (1) is found as Hence, the new system (1) is a dissipative chaotic system with a chaotic attractor.
The equilibrium points of the new chaotic system (1) are found by solving the equations From (6a) and (6b), it is apparent that x = 0 and y = 0. With this choice, (6c) is also satisfied. Hence, the complete set of equilibrium points of the system (1) is given by Since the equilibrium set S is the z-axis in R 3 , the new chaotic system (1) has line equilibrium in R 3 . Since the new chaotic system (1) has an infinite number of equilibrium points, it is said to possess hidden attractor [48].
The Kaplan-Yorke fractal dimension of the new chaotic system (1) is calculated as This gives the complexity of the new chaotic system (1) with line equilibrium. The new chaotic system (1) is invariant under the change of coordinates This shows that the new chaotic system (1) has rotation symmetry about the z-axis, which is the line of equilibrium points of the system. This is a special property of the new chaotic system (1). Consequently, every non-trivial trajectory of the system (1) possesses a twin trajectory. Figure 1 shows the 2-D phase portraits of the new chaotic system (1). Figure 4 shows the calculation of the Lyapunov exponents of the new chaotic system (1) with line equilibrium. Figures 3 and 4 show the Lyapunov spectrum and bifurcation diagram, respectively, of the system with variation of parameter a in the range a = [2, 2.6]. Periodic behavior is displayed in the specific region of the parameter 2 ≤ a ≤ 2.2. However, when a > 2.2, the system exhibits chaotic behavior. In addition, the bifurcation diagram and Lyapunov exponents of the system with respect to parameter b are shown in Figures 5 and 6. Obviously, the new chaotic system (1) exhibits periodic behavior for 1.2 ≤ b < 1.38. Behavior of the system is more complex for b ≥ 1.38, in which the system generates chaotic behavior.

III. ROUTE TO CHAOS AND COEXISTING ATTRACTORS
It is easy to see that this system has rotational symmetry with respect to the z-axis as evidenced by their invariance under the transformation from (x, y, z) into (−x, −y, z). It is well known that symmetric systems generally have coexisting attractors ( [49], [50]). Thus, in this work, we investigate such complex dynamics of system (1) by plotting forward and backward continuations for the bifurcation parameter b. The coexisting bifurcation diagrams of the state variable y is illustrated in Fig. 7, in which the forward continuation from the initial values of (0.2,0.2, 0.2) and backward continuations from the initial values of (−0.2,−0.2, 0.2). There are coexisting chaotic attractors when b = 1.4 as shown in

IV. SYNCHRONIZATION OF THE NEW CHAOTIC SYSTEMS WITH LINE EQUILIBRIUM VIA INTEGRAL SLIDING MODE CONTROL
In this section, we use active integral sliding mode control to completely synchronize a pair of identical new chaotic   systems with line equilibrium for all initial conditions. As the master system, we take the new chaotic system with line equilibrium given by where X 1 = (x 1 , y 1 , z 1 ) is the state and a, b are system parameters. VOLUME 10, 2022 As the slave system, we consider the controlled new chaotic system with line equilibrium given by where X 2 = (x 2 , y 2 , z 2 ) is the state and u = (u x , u y , u z ) is the sliding mode control to be designed. The synchronization error between new chaotic systems (10) and (11) is defined via the equation The dynamics of the synchronization error is easily obtained as follows: We use active integral sliding mode control for achieving global asymptotic synchronization between the master system (10) and slave system (11). For each error variable, we consider the integral sliding surface defined as follows: Differentiating each equation in (14), we obtain the following:     ṡ x =ė x + λ x s ẋ s y =ė y + λ y s ẏ s z =ė z + λ z s z (15) The Hurwitz stability criterion holds if we take λ x , λ y , λ z as positive constants.
Based on the exponential reaching law, we set By equating (15) and (16), we derive the following: Using the error dynamics (13), we can rewrite (17) as follows. where From (18), we get the active integral sliding mode control law as follows: The main result of this section is derived as follows. Theorem 1: The new chaotic systems with line equilibrium given by (10) and (11) are globally and asymptotically synchronized for all initial conditions X 1 (0), X 2 (0) ∈ R 3 by the active integral sliding mode control law (20) where the sliding control parameters λ i , µ i and k i (where i = 1, 2, 3) are all positive.
Proof: We use Lyapunov stability theory to establish the result. We use the quadratic Lyapunov function given by where s x , s y , s z are as defined in (15). Evidently, V is a positive definite function on R 3 . We obtain the time-derivative of V aṡ From (21) and (16), we havė Simplifying (23), we geṫ Hence,V is a negative definite function on R 3 . Thus, by Lyapunov stability theory, it follows that (s x (t), s y (t), s z (t)) → 0 as t → ∞. Hence, we conclude that (e x (t), e y (t), e z (t)) → 0 as t → ∞.
For numerical simulations, we take the parameters a and b as in the chaotic case, viz. (a, b) = (2.6, 1.5).

V. CIRCUIT REALIZATION OF THE NEW CHAOTIC SYSTEM WITH LINE EQUILIBRIUM
In this section, electronic circuit realization of the proposed chaotic system is carried out using MultiSim (Version 13.0).  The dynamics of the circuit depicted in Figure 12 is simulated in MultiSim (Version 13.0).
For circuit implementation, we rescale the state variables of the new chaotic system (1) as follows: X = 2 x, Y = 2 y and Z = 2 z. In the new coordinates (X , Y , Z ), the chaotic system (1) becomes By applying Kirchhoff's laws to the designed electronic circuit, its nonlinear equations are derived in the following VOLUME 10, 2022 form where X ,Y and Z are the voltages across the capacitors C 1 , C 2 and C 3 , respectively. Equations (26) match Equation (1) when the circuit components are selected as follows: The supplies of all active devices are ±15 Volt and the operational amplifiers TL082CD are used. The corresponding phase portraits on the MultiSIM results are shown in Fig. 13. The agreement between the numerical results ( Fig. 1) and MultiSim results ( Figure 13) shows the feasibility of the proposed line equilibrium chaotic system.

VI. FPGA-BASED CHAOTIC PSEUDO-RANDOM NUMBERS GENERATORS
Chaos is defined as nonlinear dynamic systems that are exponentially sensitive to initial conditions, deterministic, and aperiodic in the long term ( [51], [52]). Since chaotic systems exhibit aperiodic dynamic characteristic behaviour and extreme sensitivity to initial conditions and system parameters, chaos and chaotic systems have been extensively used in random number generators ( [53], [54]). Random numbers can be defined as numbers that are within a certain range, their probability of occurrence is equal to each other, and there is no specific relationship between these numbers ( [55], [56]). The generated random numbers should have good statistical properties and they cannot be predicted. Random numbers are mandatory for a wide variety of cryptographic applications [57]. Because random numbers are needed in the generation and distribution of cryptographic keys, initial vector generation, authentication protocols, prime number and password generation, protection against side channel attacks and seed generation for pseudo-random numbers generators (PRNGs) ( [58], [59]). PRNGs use the seed value obtained from some entropy source as randomness source and generate sequences that cannot be computationally distinguished from true random number generators. In addition, the streams begin to repeat after a certain period of time. So the system shows periodic feature ( [60], [61]). The advantage of PRNGs is that they are inexpensive compared to other generators. Besides, the seed entropy should be large or high to ensure unpredictability ([62]- [64]). In this case, in order to increase the security and randomness of PRNGs, chaotic systems have been added to the system as a randomness source ([65]- [70]). The chaotic systems can be implemented using analog or digital circuits. The ability to model chaotic systems at high frequencies on FPGA-based chips makes chaotic systems more attractive to use in PRNG studies ([71]- [74]). The FPGA-based chaotic PRNG designs performed in recent years and the general information about these designs are presented in Table 1.
In this part of the presented study, high-speed PRNG design has been carried out to work on FPGA chip using the novel CS with LE, which is presented to the literature. The implemented high speed PRNG unit on FPGA has been encoded in 32-bit IEEE-752-1985 floating point number standard with VHDL (Very High Speed Integrated Circuit Hardware Description Language). The designed unit has been tested using Xilinx ISE Design Tools (X-ISE-DT) program. After the test phase, for the ML605 evolution board containing VIRTEX-6 FPGA chip (device, package, speed: XC6VLX75T, FF784, −3), the Place & Route operation has been performed using the X-ISE-DT program, and after the Place& Route process, the obtained chip statistics related to FPGA resource consumption have been presented. In the last phase, 1 Mbit random number sequence obtained from high speed RPNG unit has been subjected to NIST-800-22 international statistical randomness. The results obtained from the tests have been presented. The block diagram of the novel CS with LE-based high speed PRNG unit on FPGA has been given in Figure 14.
In the first stage of the study, the novel CS with LE, which is presented to the literature in this study, is given with a set of differential equations in continuous time. Various numerical algorithms have been used in literature to model the continuous-time chaotic systems on digital platforms in discrete time. Examples of these algorithms can be given as Euler, Heun, the fourth and the fifth order Runge-Kutta and the Dormand-Prince. The more parameters used in these numerical algorithms, the longer the algorithms calculation process. In addition, if the algorithm is implemented on a digital platform, both the resource consumption increases and the operating frequency of the designed oscillator decreases. As a result, the efficiency of implementations designed using chaotic oscillator including PRNG and TRNG generally decreases. In the present study, the continuous-time novel CS with LE oscillator has been discretizated using the Euler algorithm. The mathematical model for discrete-time novel CS with LE is given in Equation (27). In the Euler discretized model of the algorithm given in the equation, the initial values of x(k), y(k) and z(k) are taken as x(0) = 0.2, y(0) = 0.2 and z(0) = 0.2 and system parameters are taken as a = 2.6 and b = 1.5.
Then, by utilizing the mathematical model of discrete time novel CS with LE, the chaotic oscillator unit has been coded with the 32-bit IEEE-752-1985 floating point number standard in VHDL (Very High Speed Integrated Circuit Hardware Description Language). Units such as the extractor and the multiplier used in the design have been created using Xilinx IP-Core generator and used in the design with VHDL. As can be observed from the block diagram of the design, the unit has two 1-bit length inputs as, Run and Clk. The Clk signal is used to synchronize subunits within the unit. Indeed, the Clk signal is sent to all units synchronously. However, this signal is only shown at the input to avoid complexity in the block diagram. On the other hand, Run signal is the start signal required for system operation. When Run signal, which equals to '1', is sent to the novel CS with LE-based high speed PRNG unit, this signal comes to MUX unit. The purpose of the MUX unit is to send the initial conditions to the Core unit. Core unit is designed to calculate the expression given in Equation (28) on FPGA.
When x(k), y(k) and z(k) values are calculated in the Core unit, Ready1 = '1' and these values are transmitted to the Multiplier unit. Here, h is multiplied by each of the x(k), y(k) and z(k) values and at this moment Ready2 = '1'. VOLUME 10, 2022 ). (29) x(k), y(k) and z(k) values from the MUX unit and S1, S2 and S3 values are entered into the adder unit. Here, the calculation of Equation (29) is performed by summing S1 signal with x(k), S2 signal with y(k) and S3 signal with z(k), respectively. The values calculated here are sent to the Filter unit. The function of the filter unit is to filter the unwanted signals generated by the novel CS with LE-based oscillator unit.
The initial conditions have been sent to the oscillator by MUX, before the oscillator produces its first results. As the novel CS with LE-based oscillator unit starts to produce its first results, the MUX unit sends these results produced by the the novel CS with LE-based oscillator unit to the input of the oscillator unit as the initial condition. The novel CS with LE-based oscillator unit generates its first results at the end of 57 clock cycle by using the initial conditions taken from the MUX unit. 3 32-bit signals namely, x out , y out and z out receiving from the oscillator unit and 3 1-bit signals as, x ready , y ready and z ready which indicate that x out , y out and z out   Bit), which is the most variable part of the 32-bit number receiving from oscillator unit, and send these bits to the MUX unit. Counter unit is a counter that is connected to the MUX unit and operates as 00-01-10. Whenever the value of the counter changes, the MUX unit receives the numbers from the 3 parallel outputs in a fragmented form respectively and transfers them to the PRNG output. PRNG structures presented in the literature generally consist of three main parts: oscillator part, quantification part and post/art processing part. Post Processing is generally used for the numbers generated by PRNG to successfully pass the NIST-800-22 statistical randomness tests. If Post Processing is used, the bit generation rate of PRNG decreases depending on the Post Processing structure used in PRNG design. In this presented study, apart from the PRNG structures presented to the literature, a high-speed PRNG unit design, that successfully passes randomness tests without the need of using Post Processing unit, is presented. A test bench unit has been coded in VHDL in order to test the design in X-ISE-DT program.
In Figure 15, simulation results of A Novel CS with LE-based high speed PRNG unit designed on FPGA using X-ISE-DT are given. As can be seen from the results, the A Novel CS with LE-based high speed PRNG unit on the FPGA can produce a result with every clock pulse. In Fig. 15, the simulation results of the novel CS with LE-based high speed PRNG unit designed on FPGA are given using X-ISE-DT. As can be seen from the results, the novel CS with LE-based high speed PRNG unit on the FPGA can generate the iterated results for each clock pulse after producing the first results. In order for the randomness tests to be successful, P-values should be equal or greater than 0.01 (P-values ≥0.01). In this case, the related test is considered as successful. In order for the generated number sequence to be considered as random, it should pass all tests in the NIST-800-22. Table 3 shows the NIST-800-22 test results for the novel CS with LE-based PRNG unit on FPGA. As can be seen in Table 3, the novel CS with LE based PRNG unit on FPGA has successfully passed all NIST-800-22 tests.

VII. IMAGE ENCRYPTION
Chaotic systems have many applications in image encryption ( [42], [77]- [80]). Bao et al. [77] proposed a discrete memristive neuron model and discussed its interspike interval-encoded application in image encryption. Wang et al. [78] proposed an effective sine modular arithmetic chaotic model and applied it to image encryption. Lai et al. [42] presented a new memristive neuron hyperchaotic model and proposed a new encryption scheme to apply the memristive neuron to the application of image encryption. Xian et al. [79] proposed a new spatiotemporal chaotic system and constructed a new cryptographic system to implement permutation-diffusion synchronous encryption. Zhu et al. [80] proposed a new sinusoidal-polynomial composite chaotic system and applied it to build a new image encryption scheme.
In this section, the proposed a novel 3D chaotic system with line equilibrium is applied for image encryption to resist statistical attack, brute force attack and differential attack. The image encryption system mainly includes pixel-level scrambling, bit-level scrambling, and pixel value diffusion.
The encryption algorithm not only shuffles the pixel positions of the image, but also replaces the pixel values with different values, which can effectively resist various attacks such as brute force attack and differential attack. The flow of image encryption system is shown in Figure 16.

A. THE PROPOSED CRYPTOSYSTEM 1) KEY GENERATION
Using SHA-256 algorithm, a 256-bit binary hash value H is obtained by encrypting plain image, which is used to generate the initial values of 3D chaotic system. H is divided into 32 bytes by byte, denoted as: h 1 , h 2 , . . . , h 32 . According to formula (30), the initial values of the 3D chaotic system x 0 , y 0 and z 0 are obtained. The initial values generated in this way has the advantage of randomness. where x 0 , y 0 and z 0 are given values. Then, the proposed 3D chaotic system will produce three pseudo-random sequences, denoted as sequence LX , LY and LZ , respectively.

2) PIXEL SCRAMBLING
Image scrambling is to destroy the correlation of adjacent pixels by rearrangement of pixel positions, so as to achieve the purpose of information confusion and secure image transmission. Given a 2D image matrix P with the size M * N , image scrambling is mainly to find a 2D scrambling matrix Q, and the 2D matrix P' can be obtained after the transformation of the matrix P through the matrix Q. Then the corresponding relationship between P and P' as follows: where Given the pseudo-random sequence LX , the sequence is arranged in ascending order to obtain the permutation index sequence IX . The scrambling matrix Q can be obtained by filling IX with M values in each row, each element T i,j in T corresponds to each element IX k in IX as follows:

3) BIT SCRAMBLING
Josephus is a cyclic traversal problem, which is described as follows: there are n people sitting around a round table, numbered from 1 to n. Now start counting from the person whose number is s, and the person who counts to r is removed, and then count again from the next person listed, and the person who counts to r is removed again. Repeat this until all the people are removed. For any given n, s and r, the order in which the n people are removed can be obtained, the order is a Josephus sequence, and this process is called Josephus traversal. If the outgoing order is regarded as a traversal sequence, it is called Josephus traversal. The Josephus traversal function is defined as: J = f (T , n, s, r), T stores the initial sequence before the function execution, and J stores the sequence after the traversal. n is the length of the sequence T , s is the starting position of the traversal (1 ≤ s ≤ n), and r is the interval value of traversal (1 ≤ r ≤ n).
In this paper, Josephus traversal is combined with chaotic sequence, and the pseudo-random sequence generated by 3D chaotic system is used as the starting positions of Josephus traversal to scramble the binary sequence of pixel. For each pixel, a different starting position is used. The pseudo random sequence LY generated by 3D chaotic system is processed according to the equation (33) to obtain the sequence LY , which is transformed into a matrix of the same size as the image. This matrix will be used for the starting position of Josephus traversal for image pixels.

4) DIFFUSION USING BIT MANIPULATION
Good diffusion method can transform the plain image into the encrypted image with uniform distribution of pixel values, which is not vulnerable to statistical attack. In this paper, the pseudo-random sequence LZ generated by 3D chaotic system is used to perform bit-XOR operation on pixel values. The LZ sequence was preprocessed according to equation (34).

5) THE PROPOSED ALGORITHM
The steps of the proposed algorithm are as follows: Step 1: • Transform plain image P into a 2D matrix P 1 with size M * N ; • According to the plain image, the hash value is obtained by using SHA-256 algorithm, and the initial values x 0 , y 0 , z 0 of 3D chaotic system are generated; • Iterating 3D chaotic system, discard a certain number of iteration values to remove the transient effect, and three sequences LX , LY , and LZ with length of M * N are obtained; • Sort the sequence LX in ascending order, generate the index sequence and convert it into matrix form, and use the index matrix to scramble the pixel positions of the image; and get the scrambled image matrix P 2 .
• Preprocess the sequence LY according to equation (33), and convert it into matrix form, denoted as sequence matrix MY , as the starting positions of Josephus traversal, scrambling each pixel in bit-level to obtain the scrambled image matrix P 3 .
• Preprocess the sequence LZ according to equation (34), and the image is diffused according to equation (35) to obtain the image matrix P 4 , which is the cipher image.

B. EXPERIMENTAL RESULTS AND ANALYSIS
To test the feasibility and effectiveness of the cryptosystem, four different test images (Lena, Baboon, Boat and Pepper) are taken. Given the values x 0 = y 0 = z 0 = 0.01, the experimental results are shown in Fig. 17. All images used in this experiment are 256 * 256 pixels in size. To prove the security of the encryption algorithm, the security analysis is carried out from the aspects of the histogram, correlation and information entropy.

1) HISTOGRAM ANALYSIS
Histogram shows the distribution of image pixel values. Histogram analysis of image can reflect the ability of encryption algorithm to resist statistical attack in terms of scrambling and diffusion. The histogram of plain image has a certain statistical rule, while cipher image should provide as little statistical information as possible. Fig. 18 shows the histograms of Lena image and its cipher image. It can be clearly seen from the figure that the pixel values of the plain image are distributed in a concentrated and regular manner, and the pixel values of the corresponding cipher image are uniform distributed, so it is difficult for attackers to obtain valuable statistical information from cipher image.

2) CORRELATION ANALYSIS
Pearson correlation coefficient is used to quantify the correlation degree of two adjacent pixels of an image, and its calculation formula is shown in (36): where (x i , y i ) is a pair of adjacent pixels, the adjacent direction can be horizontal, vertical or diagonal. N is the number of randomly selected adjacent pixel pairs.x is the average of all x i andȳ is the average of all y i . In this paper, the correlation between adjacent pixels of different plain images and their cipher images is analyzed. The results are shown in Table 4. The distributions of correlations for lena image are shown in Fig. 19. The results show that the algorithm can effectively reduce the correlation between adjacent pixel values.

3) INFORMATION ENTROPY ANALYSIS
Information entropy reflects the chaotic degree of a system, and its calculation method is shown in formula (37): Here, h(i) represents the probability of the grayscale value i. The gray level of test images in this paper is L = 256. The theoretical value for an cipher image is 8. The experimental data are shown in Table 5. The information entropy of the   plain image is small, and the information entropy of the cipher image is very close to 8.

VIII. CONCLUSION
In this work, we introduced a new chaotic system with line equilibrium. The proposed system with an absolute function nonlinearity has rich dynamics as confirmed by various attractors, Lyapunov exponents and bifurcation diagram. It is multistable as it generates coexisting chaotic attractors. As an engineering application, we have shown new synchronization results for the new chaotic system with line equilibrium using Integral Sliding Mode Control (ISMC). In addition, the designed circuit shows the feasibility of the proposed new chaotic system with line equilibrium. Furthermore, implementation of the new chaotic system by using a field-programmable gate array (FPGA) based on Pseudo-Random Number Generators (PRNGs) was presented in this work. By using FPGA-based implementation of the new chaotic system, a novel high-speed Pseudo-Random Number Generators (PRNGs) algorithm was contructed and its implementation was detailed. Finally, an image encryption algorithm based on the pixel-level scrambling, bit-level scrambling, and pixel value diffusion is proposed. The experimental results show that the encryption algorithm not only shuffles the pixel positions of the image, but also replaces the pixel values with different values, which can effectively resist various attacks such as brute force attack and differential attack. He has also conducted many workshops on control systems and chaos theory using MATLAB and SCILAB. His current research interests include control systems, chaotic and hyperchaotic systems, backstepping control, sliding mode control, intelligent control, computational science, and robotics. 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 and coauthored a number of research articles in high impact journals. His current research interests include 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), where he is also the Head of the Fixed Point Theory and Applications Research Group, and also with the Theoretical and Computational Science Center (TaCS-Center). He is also the Director of the Computational and Applied Science for Smart Innovation Cluster (CLASSIC Research Cluster), KMUTT. His research interests include fixed point theory, variational analysis, random operator theory, optimization theory, and approximation theory. Also, fractional differential equations, differential game, entropy and quantum operators, fuzzy soft set, mathematical modeling for fluid dynamics and areas of interest 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. He has provided and developed many mathematical tools in his fields productively over the past years. He has over 800 scientific papers and projects either presented or published. Moreover, he is the editorial board of more than 50 journals and also he delivers many invited talks on different international conferences every year all around the world.