Adaptive Sliding-Mode Synchronization of the Memristor-Based Sixth-Order Uncertain Chaotic System and Its Application in Image Encryption

This article presents a memristor-based sixth-order chaotic circuit which is designed based on Chua¡ ¯ s circuit using flux-controlled memristors and charge-controlled memristors. The chaotic characteristics are analyzed, and the chaotic phase diagrams are drawn. The specific upper bound information of the model uncertainty and external disturbance is unknown. We design an adaptive terminal sliding-mode control law for such chaotic systems, which not only compensates the influence of the uncertainty and disturbance but also ensures that the synchronization error system is fixed-time stable when the sliding motion takes place. Also, the accessibility of the sliding surface is guaranteed. Thus, the sufficient conditions for the synchronization of the considered systems are derived. Simulation examples show the significance and superiority of the control scheme. The synchronization strategy is applied to image encryption, and the results show that the encryption effect is excellent and has strong anti-disturbance ability.


INTRODUCTION
Chaos theory, one of the greatest scientific studies in the 20th century, establishes a link between determinism and probability theory and shows that even the simplest deterministic system also had randomness. Lorenz's weather models [1] showed that small differences in the initial conditions of a system could make a tremendous effect on the final phenomenon. So, chaos was considered one of the most creative revolutions in the scientific world [2][3][4][5][6]. Lai et al. [7] presented the latest research progress in the field of chaotic systems. The memristor is a kind of nonlinear two-terminal device that expresses the relationship between the magnetic flux and charge. Compared with resistances, capacitances, and inductances, memristors have memory function and complex nonlinear characteristics [8], so they are an ideal component for designing complex nonlinear systems. Sun et al. [9][10][11][12] applied various memristors to the Pavlov associative memristor neural network circuit for enhancing the memory and learning ability of the circuit, which made progress in the research field of brain-like systems. Li et al. [13,14] constructed a four-order memristor chaotic oscillator, which increased the amplitude and frequency of the output waveform and realized an analog circuit consistent with the theory. If it is introduced into chaos, the chaotic system will indicate more complex chaotic characteristics. In [15], the piecewise magnetically controlled memristor was used for the first time to replace Chua's diode, and the memristor chaotic circuit was designed to obtain the third-order memristor chaotic system model, and the chaotic characteristics were analyzed. The results showed abundant chaotic characteristics.
At present, many scholars have improved the chaotic model with memristors [16,17]. In [18], the three-dimensional Lorentzian system was improved to construct a new memory chaotic system. In [19], a new flux-controlled memristor with a hyperbolic cosine function for Chua's circuit was designed. In [20], the adaptive projective synchronization of three chaotic systems with three unknown parameters was studied. However, their models are all chaotic systems of fifth-order or less, and there is little research on sixth-order chaotic systems. Compared with traditional chaotic circuits, memristor chaotic circuits have stronger randomness, which are more widely used. Hence, the first innovation of this article is to construct a novel memristor-based six-order chaotic system. At the same time, we can also find that it has great significance in solving the design and synchronization problems of high-dimensional chaotic systems.
Chaotic synchronization [21,22] is an important research hotspot in complex system control and communication. In recent years, sliding-mode synchronization has attracted the attention of researchers. The sliding-mode control [23,24] forces the synchronization error systems to move along the sliding surface. Adaptive control [25] could use the running data to make the estimated value closer to the unknown quantity, to solve the influence of internal uncertainty and external disturbance during the system operation. In [26], sliding-mode controllers and adaptive laws were designed to solve the transmission synchronization problem. In [27], a global nonlinear integral sliding-mode controller was designed, and the simulation results showed that the error convergence speed was improved. In [28], the synchronization problem of a class of nonlinear chaotic systems with mismatched disturbances was studied, and a new nonsingular terminal sliding surface was designed. In addition, the traditional asymptotic synchronization time range is infinite, which does not meet the actual requirements.
Finite-time chaos synchronization [29] has attracted more and more interest due to its dynamic properties, which can reduce the cost, improve the robustness, and achieve fast convergence. In [30], a controller to satisfy the finite-time synchronization problem of chaotic systems was designed. In [31], a multi-switching rule was used to achieve finite-time hybrid synchronization for multisystems. In [32], a finite-time continuous controller was designed to reduce the control cost of nonlinear systems. In addition, the setting time of the fixed-time synchronization is independent of the initial value but is affected by the system model and parameters, which is better than finite-time hybrid synchronization. In [33], the fixed-time synchronization was realized for the synchronization problem of the BAM neural network. Xu et al. [34] studied the finite-time and fixed-time synchronization of inertial neural networks and proved the difference between the two methods through simulation results. Wang et al. [35] realized the fixedtime and finite-time synchronization of the memristor chaotic system with sliding-mode control, and the superiority of fixedtime was proved by simulation. In summary, we will design an adaptive sliding-mode controller to realize the fixed-time synchronization of the new chaotic system.
For the application research of chaotic synchronization, it is known that the unpredictability of initial conditions made chaotic systems used in the field of image encryption [36,37]. Lai et al. [38,39] constructed and studied the image processing of non-equilibrium chaotic systems and memristor chaotic systems. In [40], the Lorenz system with an additional Fourier model was applied to the raster image encryption. In [41], a seventh-order memristor chaotic system was constructed, and a new and more complex algorithm was designed, which greatly enhances the effect of pixel encryption. In [42], a new four-dimensional chaotic system was constructed and applied to the field of image encryption. Then, we can use the theoretical results obtained in this article to apply the system to the application research of image encryption. This is the application innovation of this article. Event-triggered control [43][44][45] can save network resources and reduce energy consumption, which will become our next important research direction.
Inspired by the aforementioned contents, the contributions lie in the following three aspects: 1) A memristor-based sixth-order chaotic circuit is achieved by using flux-controlled memristors and charge-controlled memristors. Different from the third-order and fourthorder memristor chaotic systems described previously, this system has a higher order and a more complex model structure. The system model is analyzed by Kirchhoff's law, and the chaotic characteristics and stability are analyzed. 2) A terminal sliding-mode surface and adaptive law are designed to solve the internal uncertainty and external disturbance. Different from the finite-time synchronization described previously, we realize the fixed-time synchronization scheme between sixth-order chaotic systems.
3) The synchronization method designed is applied to the image encryption algorithm based on obfuscation diffusion, and the image encryption process is realized. The results show that the better the encryption result, the higher the degree of antidecryption.

MEMRISTOR-BASED SIXTH-ORDER CHAOTIC CIRCUITS AND THEIR DYNAMICS ANALYSIS
A memristor-based chaotic circuit based on Chua's circuit is designed using two flux-controlled memristors and one chargecontrolled memristor. The circuit diagram is described as follows: The flux-controlled memristor model used in this study is as follows: where q and φ are the charges and magnetic fluxs, respectively, k and l are constants, and W(φ) is conductance. Also, the charge-controlled memristor [46] used in this article is as follows: where M(q) represents the resistance value. dq dt represents the internal state. i M is the current flowing through the resistor, and m, n, α, and β are constants.
Remark 1. Compared with the aforementioned thirdorder and fourth-order memristor chaotic systems, our system is extended to the sixth-order, which has a more complex structure, and the three memristors make the model obtain rich chaotic dynamic characteristics.
In light of Kirchhoff's voltage and current law analysis in Figure 1, the following equations can be obtained: where i L (t) is the current of L, V 1 (t) and V 2 (t) are the voltages of C 1 and C 2 , respectively, φ 1 (t), φ 1 (t), and q 1 (t) are the magnetic fluxes and charges of the three memristors, After normalization, the system equation is as follows: The next step is to analyze the dynamic characteristics. First, we investigate the chaotic characteristics. We choose a = 7, b = 1, c = 12, α = 0.3, β = 0.3, n 1 = −1.2, n 2 = 0.9, n 3 = 0.03, m 1 = 3.1, m 2 = 3.0, m 3 = 0.12, and then Figure 2 shows the state variables of Eq. 2. Figures 3-6 are the chaotic phase diagrams of 6 , which show the double vortex chaotic attractor.Second, the dissipation value of the calculated system is as follows: The dissipation value of the system is less than zero, so the system is dissipative. Then, we calculate the following Lyapunov exponents of the memristor-based sixth-order chaotic L 1 = 0.2733, L 2 = 3.5526, L 3 = -9.4124, L 4 = -0.0071, L 5 = -0.0454, L 6 = -6.0755. Then, it can be found that two Lyapunov indices are greater than 0. Referring to the theory of [47], the system is a chaotic system.
The next step is to analyze the stability of the system. The equilibrium point of Eq. 4 can be determined by the solution of Eq. 6, The equilibrium solution is where γ and ε are constants. It shows that the chaotic system has a plane equilibrium point corresponding to x 4 and x 5 . The Jacobian matrix of the balanced set is shown as follows: The characteristic equation is obtained by the Jacobian matrix Eq. 7 as follows: There are always two zero eigenvalues, and the other eigenvalues are determined by the system parameters. Because c, a, n 1 , n 3 are positive, and λ 3 is greater than zero. According to the stability theory of chaotic systems, the aforementioned memristor-based chaotic system is unstable at such an equilibrium point.

DESIGN OF THE ADAPTIVE SLIDING-MODE SYNCHRONIZATION SCHEME
First, the drive system is Eq. 4, and the response system Eq. 9 is as follows: where ϕ i (t) is the uncertain unknown model in Eq. 9, and f i (t) is the external disturbance, and |ϕ i (t)| ≤ ψ i , |f i (t)| ≤ ω i , ψ i and ω i are constants, and u i (t) is the control input. Therefore, the error dynamic system can be expressed as: The next step is to design the synchronization controller u i to realize the stability of the aforementioned error dynamic system. The sliding-mode control method is used to achieve the adaptive synchronization of memristor-based chaotic systems. The following lemmas will be used: Lemma 1. [36] Consider the system _ x f(x, t), when the continuous bounded functions V(x, t) satisfy the following conditions: where a > 0, b > 0, c > 0, 0 < χ < 1, δ > 1, and the system will achieve fixed-time stability and Lemma 2. [48] For any real numbers ] 1 , ] 2 , . . ., ] n and 0 < g 1 < 1, 1 < g 2 , it satisfies that Then, we select the following sliding surface as: where 0 < v < b, 0 < m < n, and j, k, l > 0, i = 1, . . ., 6, and we have ). (14) The adaptive sliding-mode controller is designated as: where ρ, k are positive constants, andψ i ,ω i are estimates of ψ i , ω i . Also, the adaptive lawsψ i ,ω i are selected as: Theorem 1. Under the influence of the controllers Eq. 13 and the adaptive laws Eq. 15, the error system Eq. 10 will reach sliding surfaces.
Proof. The following Lyapunov function is constructed: Frontiers in Physics | www.frontiersin.org April 2022 | Volume 10 | Article 863668 Taking the derivative of Eq. 17, one has Substituting Eq. 14 into the aforementioned equation, we have | sgn e i t ( ) ( ) Furthermore, we take the control scheme Eq. 13 into Eq. 10 and Eq. 18, we get According to the adaptive laws of Eq. 15, one has For the inequality (22), _ V i (t) 0 if and only if _ F i (t) 0; otherwise, _ V i (t) < 0. According to the Lyapunov stability principle, the error system can move to the sliding surface.Theorem 2. After the error system Eq. 10 reaches the sliding surface, then the state variables of Eq. 10 will tend to zero in a fixed-time T 1 , which means there will be fixed-time synchronization between the two chaotic systems Eqs 4, 9.
Proof. The following Lyapunov function is constructed: When the system reaches the sliding surface, it will satisfy F i (t) 0, _ F i (t) 0, i 1, . . . , 6, that is, (25) Because the error systems satisfy Therefore, According to Lemma 2, we continue to calculate Eq. 26 as According to Lemma 1, we get that the error state variables of Eq. 10 will tend to zero in a fixed-time. Also, using Eq. 27, the setting time is Then, the proof of Theorem 2 is completed, which means that fixed-time synchronization is reached between the two chaotic systems Eqs 4,9. Remark 2. Different from finite-time synchronization control, the bounds of fixed-time synchronization discussed in this article are only affected by system parameters, which are independent of the values of the initial time of the system.
From Figures 7, 8, the response time of the error systems and state variables is less than 1.2s under the adaptive sliding-mode controller Eq. 15 and the adaptive laws (16). From Figures 9, 10, update parameters m and n converge to fixed values at 0.82 s. The simulation results, show the superiority of the designed adaptive sliding-mode controller. Our next step is to apply the simulation results to image encryption. Remark 3. Different from the simulation results of finite-time synchronization control, the memristor-based sixth-order chaotic systems achieve fixed-time synchronization, which has a shorter stable time and whose setting time only depends on our system's parameters.

APPLICATIONS IN IMAGE ENCRYPTION
Because the memristor chaotic circuit has stronger noise like initial value sensitivity and long-term unpredictability, the application of the memristor-based chaotic circuit to image encryption will make the experimental images have a stronger anti-decoding function. We verify the effectiveness of the adaptive sliding-mode synchronization scheme in image encryption by using a chaotic obfuscation-diffusion mechanism [49], and the specific implementation process is shown in Figure 11.According to Figure 11, we first read three data matrices, R, G, and B, of the original image. Three parameters are selected from the driving system (4), x 2 , x 3, and x 4 are selected from the memristor-based chaotic system designed in this article, which is to do the ascending and descending sequence and row displacement of R, G, B. We can obtain the data of R′, G′, and B′. Next, the pixel value is replaced by x 5 to get R″, G″, and B″. We choose the following parameter as: The intermediate variable Q is as follows: where z and x are pixel row and row position, S is gray value. We get the matrix of the encrypted image R″, G″, B″ as: The binary sequence R″, G″, B″ is converted into a twodimensional matrix R′′′, G′′′, B′′′, and the two-dimensional matrix R′′′, G′′′, B′′′ is reconstructed to obtain the encrypted image.   The above process belongs to the image encryption process. For the image decryption process, we read the three color data matrices R′′′, G′′′, B′′′ of the encrypted image. Then the pixel value is replaced by y 5 to get R″, G″, B″. And we set n′ zx abs y 5 z, x ( ) − round y 5 z, x ( ) × 10 2 m′ zx abs y 5 z, x ( )− round y 5 z, x ( ) × 10 3 Then, we can easily get From the aforementioned equations, we can get The binary sequence R″, G″, B″ is converted into a twodimensional matrix R′, G′, B′. Three parameters are selected from the response system Eq. 9, y 2 , y 3 , y 4 are selected to do the ascending and descending sequence and row displacement of R′, G′, B′ to R, G, B, which is reconstructed to obtain the decryption images. The memristor-based six-order chaotic circuit and adaptive sliding-mode synchronization scheme are used to make the initial value and synchronization simulation, and the  Frontiers in Physics | www.frontiersin.org April 2022 | Volume 10 | Article 863668 9 image encryption experiment is carried out. We selected two images for the encryption experiment, and the experimental results are shown in Figure 12.
We use histogram analysis, which is an important tool to assess encryption performance, to analyze Figure 12. The experimental results are shown in Figure 13.Then, we use chi-square test Eq. 38 to analyze the encrypted images' histograms in Figure 13, where f q is the frequency distribution of the encrypted images' histograms and n = (α*β)/256, for a α*β dimension image. We get 268.2371. Compared with x 2 0.05 (255) 293.24783, x 2 image01 , x 2 image02 < x 2 0.05 (255). It can be seen that the original images' histograms fluctuate greatly, while the encrypted images' histograms are evenly distributed and have fluctuation. A good encryption algorithm can make the distribution of encrypted images as uniform as possible. Therefore, the adaptive sliding-mode controller makes the memristor-based sixth-order chaotic system suitable for image encryption, which improves the security of signal transmission.

CONCLUSION
This article has established a memristor-based sixth-order chaotic circuit. The dynamic analysis and sliding-mode synchronization and its application have been studied. A sixth-order chaotic system has been constructed by using flux-controlled memristors and charge-controlled memristors. For the synchronization of sixth-order memristor chaotic systems with unknown models and external disturbances, a class of terminal sliding-mode surface and adaptive control laws is designed. After coming to the sliding-mode surface by asymptotic stabilization, the system achieves synchronization in a fixed time, and a simulation example shows the effectiveness of the scheme. Furthermore, the sixth-order chaotic system and synchronization scheme have been applied to image protection. The results show that the algorithm can be combined with the system, which has a good encryption effect and high anti-interference. This work applies the proposed memristor-based sixth-order chaotic system as a network node to the synchronization of complex networks.