A method for constructing, dynamic analysis and application of a six-dimensional memristive hyperchaotic system

Abstract

The application relates to encryption communication technology, and particularly discloses a construction, dynamic analysis and application method of a six-dimensional memristor hyperchaotic system. The method discovers that the new system has unique dynamic behaviors through a phase diagram, a bifurcation diagram and a Lyapunov exponent spectrum, generates symmetric coexistence cycles, chaos and hyperchaos double-wing attractors which depend on system parameters, shows extremely strong initial value sensitivity, has special symmetric extreme multistability and a memristor initial value enhancement behavior, generates symmetric four-wing attractors by adding a constant controller, shows more complexity and diversity, and finally, simulation is carried out through Multisim, the generated attractor is consistent with a phase diagram obtained through numerical simulation, and the feasibility of the new system is verified. The application has great advantages in encryption security and attack resistance.

Description

TECHNICAL FIELD

[0001] The application relates to the field of encrypted communication, and in particular to a construction, dynamic analysis and application method of a six-dimensional memristor hyperchaotic system. BACKGROUND

[0002] A memristor is a new type of circuit device with low power consumption and easy integration, and is widely used in electronic circuits and related fields. Research shows that the introduction of a memristor into a chaotic circuit can discover complex dynamic behaviors, such as symmetric or asymmetric coexisting attractors, extreme multistability and hidden coexisting attractors. In 2021, a new memristor was proposed by those skilled in the art and introduced into a chaotic circuit, and it was determined that the system has a hidden chaotic attractor, and the existence of coexisting attractors was analyzed. In the same year, those skilled in the art also proposed a four-dimensional double-wing chaotic system that can produce transient chaos, sustained chaos and bistability. In the following year, those skilled in the art introduced an absolute value memristor as a feedback item of the VB19 chaotic system, and proposed a simple four-dimensional memristor chaotic system. The six-dimensional memristor hyperchaotic system can be locally adjusted in amplitude and can produce coexisting attractors. At the same time, those skilled in the art also proposed a new three-dimensional conservative system, which not only has rich symmetry, but also discovered the initial value enhancement behavior of the memristor.

[0003] At present, although those skilled in the art have proposed a series of memristor hyperchaotic systems with complex dynamic phenomena, due to the complexity, high sensitivity and numerical calculation of high-dimensional systems, these memristor hyperchaotic systems with complex dynamic phenomena are mainly based on low-dimensional systems, and there is no research on higher-dimensional memristor hyperchaotic systems.

[0004] In recent years, chaotic systems with multi-stability have attracted the attention of many scholars. Such systems can produce multiple stable states coexisting attractors, showing great flexibility. When subjected to external disturbances, different states can be switched to ensure the normal operation of the system. Therefore, the study of multi-stable systems is of certain significance to image encryption, information engineering and other fields. Those skilled in the art have constructed a series of chaotic systems with multi-stability. For example: Vijayakumar et al. proposed a four-dimensional hyper-jerk chaotic system with a single equilibrium point, which can observe multiple types of coexisting attractors, showing multi-stable characteristics. In 2022, those skilled in the art constructed a four-dimensional smooth Chua system and found chaotic attractors, periodic or quasi-periodic coexisting attractors, etc., showing unique multi-stable phenomena. On the basis of multi-stability, those skilled in the art have constructed a memristor chaotic system with extreme multi-stability. For example: In 2020, Chang et al. constructed a four-dimensional memristor chaotic system, described the generation mechanism, characteristics and stability analysis of the system, and discussed the hidden symmetric extreme multi-stability and complex dynamic characteristics. The next year, Hou et al. introduced a memristor into a tabu learning neuron model and observed special single-period and multi-period infinite coexisting attractors.

[0005] Although those skilled in the art have proposed a series of memristor chaotic systems with multi-stability, the above-mentioned memristor chaotic systems with multi-stability do not exhibit symmetric extreme multi-stability and fail to produce unique memristor initial value enhancement behavior. Therefore, a new six-dimensional memristor hyperchaotic system is proposed, and it is necessary to further analyze the special dynamic characteristics of high-dimensional memristor chaotic systems. SUMMARY

[0006] The purpose of the present application is to solve the above problems, and a six-dimensional memristor hyperchaotic system construction, dynamic analysis and application method is designed.

[0007] To achieve the above-mentioned purpose, the technical scheme of the present application is a construction method of a six-dimensional memristor hyperchaotic system. The construction method is based on the dynamic phenomena of hyperchaotic systems to construct a model of a six-dimensional memristor hyperchaotic system and analyze the stability of the model of the six-dimensional memristor hyperchaotic system.

[0008] The dimensionless model of the hyperchaotic system is:

[0009]

[0010] In formula (1), x, y, z, w represent four state variables, and a, b, c, d represent system parameters.

[0011] The mathematical model of the quadratic piecewise nonlinear memristor and the cubic flux-controlled memristor embedded in formula (1) is shown in formula (2) and formula (3) respectively:

[0012]

[0013]

[0014] wherein is the flux-controlled memductance, is the magnetic flux, i and v are the current and voltage flowing through the two ends respectively, and α and β are control parameters; on the basis of formula (1), by introducing two memristors W(m)=α-β|m| and W(n)=α+βn2, the coupling coefficients of the state variables w in the first equation and x in the fourth equation are replaced respectively, and the original system xz, xy and yz terms are rewritten as xz2, xyz and yz2 terms, a six-dimensional memristive hyperchaotic system is obtained, and the mathematical model is as follows:

[0015]

[0016] wherein x, y, z, w, m and n are state variables of the system, and a, b, c, d, e and f are control parameters of the system; according to the mathematical model, formula (3) has invariance under the transformation of (x, y, z, w, m, n)→(x, y, -z, w, m, n),

[0017] (x, y, z, w, m, n)→(-x, -y, z, -w, -m, -n) and (x, y, z, w, m, n)→(-x, -y, -z, -w, -m, -n), which can provide conditions for the formation of symmetric coexisting attractors.

[0018] The process of stability analysis of the model of the six-dimensional memristive hyperchaotic system is as follows: first, the right side of formula (4) is equal to 0, and formula (5) is obtained:

[0019]

[0020] The equilibrium point of the six-dimensional memristive hyperchaotic system can be obtained through formula (5), and then a face equilibrium point of the six-dimensional memristive hyperchaotic system can be obtained through numerical calculation

[0021] O={(x, y, z, w, m, n)|x=y=z=w=0, m=ε, n=δ}, wherein ε and δ are real constants, O is linearized, and the Jacobian matrix JE of the equilibrium point set O is:

[0022]

[0023] Select the system parameters a = 75, b = 70, c = 2, d = 15, e = 1, f = 2, alpha = 1, beta = 0.01, the characteristic equation of the system at the equilibrium point set O is obtained:

[0024] lambda2(lambda-d)(lambda+e)(lambda2+a*lambda+p+q) = 0 (7)

[0025] Wherein:

[0026] p = fc * beta * | epsilon | * (alpha + delta2 * beta) (8)

[0027] q = - fc * alpha * (alpha + delta2 * beta) (9)

[0028] From the characteristic equation (7), the system always exists two zero eigenvalues lambda1 = lambda2 = 0, and the remaining four non-zero eigenvalues depend on the system parameters, since the system parameters are all greater than 0, there is always a positive eigenvalue lambda3 = d and a negative eigenvalue lambda4 = -e, by Routh criterion, the system is unstable at the equilibrium point.

[0029] A dynamic analysis method for a six-dimensional memristive hyperchaotic system, characterized in that the analysis method analyzes the dynamic behavior of the six-dimensional memristive hyperchaotic system through phase diagrams, bifurcation diagrams and Lyapunov exponent spectrum.

[0030] The dynamic behavior of the six-dimensional memristive hyperchaotic system includes:

[0031] First, the coexistence symmetry dynamic characteristics of the six-dimensional memristive hyperchaotic system change with the parameter e;

[0032] Second, the six-dimensional memristive hyperchaotic system exhibits extreme multistability under different initial states of the memristor;

[0033] Third, the six-dimensional memristive hyperchaotic system exhibits memristor initial value enhancement behavior.

[0034] An application method for a six-dimensional memristive hyperchaotic system, which adds a constant controller to the model of the six-dimensional memristive hyperchaotic system, can generate a symmetric four-wing attractor, and the application method can also perform image encryption based on the six-dimensional memristive hyperchaotic system.

[0035] The application method adds a constant controller g in equation (4) to obtain a new system model, that is, equation (10), as follows:

[0036]

[0037] Select the parameters a = 75, b = 70, c = 2, d = 15, e = 10, f = 1, the initial state is X1 = (1, 1, 1, 1, 1, 1) and X3 = (-1, -1, 1, -1, -1, -1), when g = -0.8, four-winged attractors are successfully generated, and the two attractors are symmetric about the z-axis.

[0038] The application method is based on the process of encrypting pictures by the six-dimensional memristor hyperchaotic system as follows:

[0039] Step one: generation of chaotic sequence

[0040] Set parameters a = 75, b = 70, c = 2, d = 15, e = 11, f = 1, alpha = 1, beta = 0.01, time step T = 0.0005, and continuously iterate the six-dimensional memristor hyperchaotic system to generate hyperchaotic sequences (x(i), y(i), z(i), w(i), m(i), n(i), p, q);

[0041] Step two: divide the colored Lena image P into P r , P g and P b three parts;

[0042] Step three: generate four chaotic matrices S1, S2, S3, S4, S5, S6 with size mxn by the six-dimensional memristor hyperchaotic system, use the chaotic sequence S1 to perform a scrambling operation on the input matrix P, and u1 and v1 represent the permutation positions of the elements P(i,j) of the matrix P, that is, each element P(i,j) calculates a new position, and exchanges P(i,j) with P(u1,v1), thereby realizing the scrambling operation:

[0043]

[0044] Wherein, floor is the floor operation, m and n are the number of rows and columns of the matrix P respectively, rows and cols are the row vector and column vector of the matrix S1 respectively, and S1(i,j) represents the element in the i-th row and j-th column of the matrix S1, through the two formulas, each element P(i,j) in the original matrix P can be mapped to a new position (u1,v1) and permuted;

[0045] Global scrambling of P r , P g and P b is three sequences:

[0046]

[0047] Step four: the diffusion operation is realized by XOR operation on the previous two pixel values, the change of the current pixel value is sufficient, the output matrix C is calculated using the input matrix P and the chaotic sequence S, and the matrix T of each row is calculated i As:

[0048]

[0049] The output matrix C of each column is calculated i As:

[0050]

[0051] Wherein, F is the pixel value of the gray image, F=256, Indicates the down rounding operation.

[0052] Compared with the prior art, the present application has the following beneficial effects:

[0053] 1. The present application constructs a six-dimensional memristive hyperchaotic system with one face equilibrium point, reveals the coexistence bifurcation mode of high-dimensional memristive chaotic system, and finds multiple types of symmetric coexistence attractors depending on system parameters;

[0054] 2. The present application analyzes the multi-stable phenomenon of the new system, observes the unique symmetric extreme multi-stable phenomenon depending on the initial value of the memristor, and also finds the special initial value enhancement behavior of the memristor;

[0055] 3. The present application introduces a constant controller in the new system, constructs a symmetric four-wing attractor, and also encrypts the picture through the six-dimensional memristive hyperchaotic system to improve the confidentiality of data transmission. DETAILED DESCRIPTION

[0056] Figure 1 It is the dynamic phenomenon diagram of the six-dimensional memristive hyperchaotic system of the present application with parameter e, wherein (a) is the coexistence bifurcation diagram of X1 and X2; (b) is the coexistence bifurcation diagram of X3 and X4; (c) is the Lyapunov exponent spectrum generated from X1; (d) is the Lyapunov exponent spectrum generated from X3;

[0057] Figure 2 It is the phase diagram of the symmetric coexistence attractor of the six-dimensional memristive hyperchaotic system of the present application in the x-z plane, wherein (a) is a periodic attractor when e=3; (b) is a periodic attractor when e=3; (c) is a chaotic attractor when e=0.8; (d) is a chaotic attractor when e=0.8; (e) is a hyperchaotic attractor when e=10; (f) is a hyperchaotic attractor when e=10;

[0058] Figure 3are the dynamical phenomena diagrams of the six-dimensional memristive hyperchaotic system of the present application with different initial states, wherein (a) is the coexistence bifurcation diagram of Y1 and Y2; (b) is the coexistence bifurcation diagram of Y3 and Y4; (c) is the Lyapunov exponent spectrum generated from Y1; (d) is the Lyapunov exponent spectrum generated from Y3;

[0059] Figure 4 are the projection diagrams of the symmetric coexistence attractors of the six-dimensional memristive hyperchaotic system of the present application in m-y-z space, wherein (a) is a periodic attractor; (b) is a chaotic attractor;

[0060] Figure 5 are the infinite symmetric coexistence hyperchaotic attractors of the six-dimensional memristive hyperchaotic system of the present application in the z-m plane, wherein (a) is a coexistence attractor generated from Y1; (b) is a coexistence attractor generated from Y2; (c) is a coexistence attractor generated from Y3; (d) is a coexistence attractor generated from Y4;

[0061] Figure 6 are the memristive initial value enhancement behaviors of the six-dimensional memristive hyperchaotic system of the present application generated under different parameters, wherein (a) is when c=7; (b) is when c=8;

[0062] Figure 7 are the four-wing symmetric coexistence attractors of the six-dimensional memristive hyperchaotic system of the present application generated in different planes, wherein (a) is the x-z plane; (b) is the x-z plane; (c) is the y-z plane; (d) is the y-z plane;

[0063] Figure 8 is the circuit schematic diagram of the six-dimensional memristive hyperchaotic system of the present application;

[0064] Figure 9 are the phase diagrams of the four symmetric coexistence periodic attractors of the six-dimensional memristive hyperchaotic system of the present application in the x-z plane, wherein (a) is when the initial state is X1, (b) is when the initial state is X3, (c) is when the initial state is X2, and (d) is when the initial state is X4;

[0065] Figure 10 are the phase diagrams of the four symmetric coexistence chaotic attractors of the six-dimensional memristive hyperchaotic system of the present application in the x-z plane, wherein (a) is when the initial state is X1, (b) is when the initial state is X3, (c) is when the initial state is X2, and (d) is when the initial state is X4;

[0066] Figure 11is the phase diagram of four symmetric coexisting hyperchaotic attractors of the six-dimensional hyperchaotic system with memristors in the x-z plane, wherein (a) is the initial state X1, (b) is the initial state X3, (c) is the initial state X2, and (d) is the initial state X4;

[0067] Figure 12 is the encryption process of the six-dimensional hyperchaotic system with memristors, wherein (a) is the original picture, (b) is the ciphertext picture, and (c) is the decrypted picture;

[0068] Figure 13 is the pixel histogram of the six-dimensional hyperchaotic system with memristors, wherein (a) is the original picture, and (b) is the ciphertext picture. DETAILED DESCRIPTION

[0069] The present application will be described in detail below with reference to the accompanying drawings, as shown in Figures 1-13 ;

[0070] The dimensionless model of the hyperchaotic system is:

[0071]

[0072] In formula (1), x, y, z, and w represent four state variables, and a, b, c, and d represent system parameters.

[0073] The mathematical models of the embedded quadratic piecewise nonlinear memristor and the cubic magnetic control memristor are formula (2) and formula (3), respectively:

[0074]

[0075]

[0076] wherein is the magnetic control memristor, is the magnetic flux, i and v are the current and voltage flowing through the two ends, respectively, and α and β are control parameters.

[0077] On the basis of formula (1), by introducing two memristors W(m)=α-β|m| and W(n)=α+βn2, respectively replacing the coupling coefficients of the state variable w in the first equation and the state variable x in the fourth equation, and rewriting the original system xz, xy, and yz terms into xz2, xyz, and yz2 terms, a new chaotic system is obtained, and the mathematical model thereof is:

[0078]

[0079] where x, y, z, w, m, n are state variables of the system, a, b, c, d, e, f are control parameters of the system. According to the mathematical model, formula (3) has invariance under the transformation of (x, y, z, w, m, n)→(x, y, -z, w, m, n), (x, y, z, w, m, n)→(-x, -y, z, -w, -m, -n) and (x, y, z, w, m, n)→(-x, -y, -z, -w, -m, -n), which can provide conditions for the formation of symmetric coexistence attractors.

[0080] Let the right side of formula (4) be equal to 0, and the equilibrium point of the newly constructed six-dimensional memristive hyperchaotic system can be obtained by formula (5).

[0081]

[0082] Through numerical calculation, it can be obtained that the six-dimensional memristive hyperchaotic system has a plane equilibrium point O={(x, y, z, w, m, n)|x=y=z=w=0, m=ε, n=δ}, where ε and δ are real constants. Linearizing O, the Jacobian matrix J of the equilibrium point set O is obtained E

[0083]

[0084] When the system parameters a=75, b=70, c=2, d=15, e=1, f=2, α=1, β=0.01 are selected, the characteristic equation of the system at the equilibrium point set O is obtained:

[0085] λ 2 (λ-d)(λ+e)(λ 2 +aλ+p+q)=0 (7)

[0086] where:

[0087] p=fcβ|ε|(α+δ 2 β) (8)

[0088] q=-fcα(α+δ 2 β) (9)

[0089] According to the characteristic equation, there are always two zero eigenvalues λ1=λ2=0, and the remaining four non-zero eigenvalues depend on the system parameters. Since the system parameters are all greater than 0, there is always a positive eigenvalue λ3=d and a negative eigenvalue λ4=-e. According to the Routh criterion, the system is unstable at the equilibrium point.

[0090] ​By bifurcation diagram and corresponding Lyapunov exponent spectrum, it is found that the six-dimensional memristive hyperchaotic system has complex dynamical characteristics, and when certain conditions are selected, symmetric coexisting attractors can be generated. The present application mainly analyzes the dynamical behavior of formula (4) under the change of parameter e. By setting the parameters a = 75, b = 70, c = 2, d = 15, f = 1, α = 1, β = 0.01, and the interval of parameter e is [0, 12], and based on the symmetry invariance of the six-dimensional memristive hyperchaotic system, four initial states are selected, which are X1 = (1, 1, 1, 1, 1, 1), X2 = (1, 1, -1, 1, 1, 1), X3 = (-1, -1, 1, -1, -1, -1), and X4 = (-1, -1, -1, -1, -1, -1). By numerical simulation, the bifurcation diagram of the state variable z with the change of parameter e is drawn, as shown in Figure 1 , it can be observed that under the selected four initial states, the system generates four symmetric branches with the change of parameter e, indicating that the constructed six-dimensional memristive hyperchaotic system can generate multiple symmetric coexisting attractors. Then the Lyapunov exponent spectrum of the system is obtained by Wolf algorithm when the initial state is X1 and X3, as shown in Figure 1 (c) and Figure 1 (d), it can be observed that the generated Lyapunov exponent spectrum is the same, and with the change of parameter e, periodic, chaotic and hyperchaotic two-wing attractors can also be generated. Specifically, when parameter e ∈ [0, 0.2) and e ∈ [1.9, 3.3), the system presents periodic state; when parameter e ∈ [0.2, 1.9) and e ∈ [3.3, 6.5), the system presents chaotic state; when parameter e ∈ [6.5, 12], the system presents hyperchaotic state.

[0091] To further clarify the coexisting symmetric dynamical characteristics of formula (4) with the change of parameter e, the phase diagram of the attractor in the x-z plane is used to verify the multiple types of symmetric coexisting attractors generated by the system under the states of X1, X2, X3 and X4, as shown in Figure 2 . Among them, the periodic attractor generated when the system parameter e = 3 is shown in Figure 2 (a) and 2(b); the chaotic attractor generated when e = 0.8 is shown in Figure 2 (c) and 2(d); the hyperchaotic attractor generated when e = 10 is shown in Figure 2 (e) and 2(f). According to the phase diagram of the system, it is further revealed that the six-dimensional memristive hyperchaotic system can generate multiple states of symmetric coexisting attractors.

[0092] Therefore, the six-dimensional memristive hyperchaotic system not only shows higher initial value sensitivity, but also can generate multiple symmetric coexisting periodic, chaotic and hyperchaotic attractors, showing better encryption, which provides a better choice for the practical application in the field of information security, etc.

[0093] The six-dimensional memristive hyperchaotic system also exhibits extreme multi-stability under different initial states of the memristor. By setting the parameters a = 75, b = 70, c = 2, d = 15, e = 1, f = 5, a = 1, b = 0.01, and selecting four initial states Y1 = (1, 1, 1, 1, m(0), 1), Y2 = (1, 1, -1, 1, m(0), 1), Y3 = (-1, -1, 1, -1, -m(0), -1), and Y4 = (-1, -1, -1, -1, -m(0), -1), and setting the initial state m(0) of the memristor to vary within [-50, 50]. Figure 3 (a) and Figure 3 (b) shows the coexistence bifurcation diagrams of the system state variable z with respect to m(0) when the initial states are Y1, Y2, Y3, and Y4, Figure 3 (c) and Figure 3 (d) shows the Lyapunov exponent spectrum with respect to m(0) when the initial states are Y1 and Y3. From Figure 3 It can be found that the newly constructed system not only generates various types of attractors, but also exhibits unique symmetric extreme multi-stability as m(0) changes. More specifically, when m(0) e [-50, -38.8) U [-33.5, -23.3) U [15.3, 50), the system is in a periodic state in a large region, accompanied by chaotic phenomena in a small region; when m(0) e [-38.8, -33.5) U [18.8, 19.7), the system is in a chaotic state; when m(0) e [-23.3, 15.3), the system is in a hyperchaotic state.

[0094] Figure 4 (a) and Figure 4 (b) shows some typical phase diagrams of multi-symmetric coexisting attractors in the m-y-z space, which are generated when the initial states are Y1, Y2, Y3, and Y4, respectively, wherein, Figure 4 (a) is a symmetric coexisting periodic attractor generated by the system when m(0) = 45; Figure 4 (b) is a symmetric coexisting chaotic attractor generated by the system when m(0) = 19. Figure 5 (a) to Figure 5 (d) respectively shows the phase diagrams of the z-m plane projection of four groups of symmetric coexisting hyperchaotic attractors generated by the system under four different initial states Y1, Y2, Y3, and Y4, and the initial state m(0) of the memristor is set to -15, -8, -2, and 11 for each group.

[0095] It should be noted that the system also exhibits unique initial value enhancement behavior of the memristor. When appropriate initial states and system parameters are selected, the coexisting attractors generated by the system exhibit a shift behavior on the equilibrium plane.

[0096] Select system parameters a = 75, b = 70, c = 7, d = 15, e = 1, f = 16, α = 1, β = 0.01, and initial state Y1 = (1, 1, 1, 1, m(0), 1). Let m(0) gradually increase from -40 to 45. The attractor phase diagram generated by the system in the mn plane is as follows. Figure 6 As shown, where Figure 6 (a) When c = 7, it can be observed that as m(0) increases from -40 to 45, the resulting double-winged attractor can shift to the right side of the mn plane. A special memristor initial value enhancement behavior was also discovered; when c = 8, the attractor size becomes half that of c = 7, as shown below. Figure 6 As shown in (b), the six-dimensional memristor hyperchaotic system exhibits stronger initial value sensitivity and more complex offset characteristics, which can provide more convenience for the conversion and transmission of chaotic signals.

[0097] Compared to two-winged chaotic systems, multi-winged systems exhibit more complex planar or three-dimensional topological structures with wings distributed in various directions, meaning they can display more unique dynamic behaviors. According to current literature, there are few studies on introducing controllers to construct four-winged attractors in two-winged memristor chaotic systems.

[0098] To construct a four-winged attractor, based on equation (4), a simple controller g is added, resulting in a new equation (10), as follows:

[0099]

[0100] The system parameters are selected as a = 75, b = 70, c = 2, d = 15, e = 10, f = 1, and the initial states are X1 = (1, 1, 1, 1, 1, 1) and X3 = (-1, -1, 1, -1, -1, -1). When g = -0.8, the four-winged attractors generated from X1 and X3 are as follows: Figure 7 As shown. Among them, Figure 7 (a) and 7(b) respectively plot the projections of the quadrangular attractor generated in the initial states X1 and X3 onto the xz plane; Figure 8 (c) and (d) plot the projections of the quadruped attractor generated in initial states X1 and X3 onto the yz plane. Figure 7 It can be seen that a four-winged attractor can be successfully generated by introducing a constant controller, and the two attractors are symmetric about the z-axis.

[0101] Therefore, the constructed six-dimensional memristor hyperchaotic system can generate a four-winged attractor based on a two-winged chaotic system through a constant controller g, exhibiting more unique symmetric dynamic characteristics and providing more options for practical engineering applications.

[0102] Example

[0103] Firstly, an analog circuit is designed based on Multisim to verify the symmetric dynamic characteristics of equation (4) with the change of parameter e in the initial state of X1, X2, X3 and X4. LM741 is selected as the operational amplifier, the supply voltage is ±12V, and AD633 is selected as the analog multiplier. The time scale factor of the system is converted to τ = 100t. The parameters of equation (4) are selected as a = 75, b = 70, c = 2, d = 15, e = 10, f = 1, α = 1, β = 0.01. Equation (11) is obtained:

[0104]

[0105] According to Kirchhoff's circuit law, the circuit state equation is obtained as equation (12):

[0106]

[0107] The following element parameters can be obtained:

[0108]

[0109] The initial state of the system can be set by modifying the initial value of the capacitor, and the size of parameter e can be changed by modifying the resistance value of R7. When R7 = 33.33kΩ, 125kΩ, 10kΩ, the modified parameters e = 3, 0.8, 10, respectively, the system can generate symmetric coexisting periodic, chaotic and hyperchaotic attractors, as shown in Figure 9 , Figure 10 and Figure 11 , respectively. It can be observed that the simulation results are the same as the attractor phase diagrams shown in Figure 2 (a)-(f). The feasibility of the six-dimensional memristive hyperchaotic system is successfully verified.

[0110] Based on the proposed six-dimensional memristive hyperchaotic system, image encryption is performed to test the effect of the six-dimensional memristive hyperchaotic system in practical applications. The encryption process is shown in Figure 12 . The size of the original picture used for encryption is 512×512×3. The encryption process is as follows:

[0111] Step one: generation of chaotic sequence.

[0112] The parameters are set as a = 75, b = 70, c = 2, d = 15, e = 11, f = 1, α = 1, β = 0.01, and the time step T = 0.0005. The six-dimensional memristive hyperchaotic system is continuously iterated to generate hyperchaotic sequences (x(i), y(i), z(i), w(i), m(i), n(i), p, q).

[0113] Step two: the color Lena image P is divided into P r , P g and Pb Three parts.

[0114] Step 3: Generate four chaotic matrices S1, S2, S3, S4, S5, and S6 of size m x n using a six-dimensional memristor hyperchaotic system. Use the chaotic sequence S1 to scramble the input matrix P. u1 and v1 represent the permutation positions of elements P(i,j) in matrix P, respectively. That is, each element P(i,j) can be calculated to have a new position, and P(i,j) is swapped with P(u1,v1), thus achieving the scrambling operation.

[0115]

[0116] Here, floor represents the floor operation, m and n are the number of rows and columns of matrix P, respectively, rows and cols are the row vector and column vector of matrix S1, and S1(i,j) represents the element in the i-th row and j-th column of matrix S1. Using these two formulas, each element P(i,j) in the original matrix P can be mapped to a new position (u1,v1) and a permutation operation can be performed.

[0117] P r P g and P b The global scramble is performed into three sequences:

[0118]

[0119] Step 4: The diffusion operation changes the current pixel value by XORing the previous two pixel values. The change in pixel values ​​of the encrypted image is relatively sufficient. The output matrix C is calculated using the input matrix P and the chaotic sequence S, and the matrix T for each row is calculated. i for:

[0120]

[0121] Calculate the output matrix C for each column. i for:

[0122]

[0123] F is the pixel value of the grayscale image, F = 256. This indicates a round-down operation.

[0124] Step 5: Decryption is the reverse process of encryption.

[0125] The corresponding encryption process is as follows: Figure 12The R, G, B channel of the original image and the ciphertext image are analyzed in horizontal, vertical and diagonal line adjacent pixel correlation, and the information entropy is calculated, and the results are shown in Table 1, which is the correlation coefficient and information entropy value of the R, G, B channel of the original image and the ciphertext image. It can be observed that for the original image, the correlation coefficients of the R, G and B channels tend to 1, indicating that the pixels in the image have certain continuity and regularity, which is also reflected in the low information entropy value. For the decrypted image, the correlation coefficients of the three channels are very close to 0, indicating that there is almost no correlation between the pixels in the image, and it is difficult to find the characteristics of continuity and regularity from them, which is also reflected in the performance of the information entropy value close to the maximum value.

[0126] It can be seen that the six-dimensional memristor hyperchaotic system performs well in image encryption.

[0127]

[0128] Table 1

[0129] To further verify the encryption effect of the image, Figure 13 The R, G, B channel histogram of the original image and the ciphertext image is drawn. It can be found that the histogram of the original image is obviously uneven, which means that there are obvious regional features. The histogram of the ciphertext image is uniformly distributed, which means that after the original image is encrypted by the chaotic system, there is no obvious regional feature, and it is difficult to crack the content of the ciphertext image from the distribution, thereby enhancing the anti-attack ability. In addition, uniform distribution can also protect the regional features and structural features in the original image, preventing attackers from cracking the encryption process according to these features. Therefore, the image encryption based on the six-dimensional memristor hyperchaotic system has greater advantages in resisting attacks.

[0130] The above technical solutions only reflect the preferred technical solutions of the present application, and some changes made by the skilled in the art to some parts of the present application also reflect the principles of the present application and are within the scope of protection of the present application.

Claims

1. An application method for a six-dimensional memristor hyperchaotic system, characterized in that, This application method involves adding a constant controller to the model of a six-dimensional memristor hyperchaotic system, which can generate a symmetrical four-winged attractor. This application method can also be used for image encryption based on the six-dimensional memristor hyperchaotic system. The process of encrypting images using this application method based on a six-dimensional memristor hyperchaotic system is as follows: Step 1: Generation of chaotic sequences; Step 2: Divide the color Lena image P into P0... r P g and P b Three parts; Step 3: Generate six chaotic matrices S1, S2, S3, S4, S5, and S6 of size m x n using a six-dimensional memristor hyperchaotic system. Use the chaotic sequence S1 to scramble the input matrix P. u1 and v1 represent the permutation positions of elements P(i,j) in matrix P, respectively. Swap P(i,j) with P(u1,v1) to achieve the scrambling operation. Step 4: Calculate the output matrix C using the input matrix P and the chaotic sequence S; The six-dimensional memristor hyperchaotic system is constructed based on the dynamic phenomena of hyperchaotic systems. The dimensionless model of the hyperchaotic system is as follows: In equation (1), x, y, z, w represent four state variables, and a, b, c, d represent system parameters; The mathematical models of the quadratic piecewise nonlinear memristor and the cubic magnetically controlled memristor embedded in equation (1) are shown in equations (2) and (3), respectively: in It is magnetically controlled memory conduction. It is the magnetic flux, i and v are the current and voltage flowing into its two ends respectively, and α and β are control parameters; based on equation (1), two memristors are introduced. Replace the coupling coefficients of the state variable w in the first equation and the state variable x in the fourth equation, respectively, and rewrite the xz, xy, and yz terms of the original system as xz. 2 ,xyz,yz 2 The term yields a six-dimensional memristor hyperchaotic system, whose mathematical model is as follows: Where x, y, z, w, m, n are the system's state variables, and a, b, c, d, e, f are the system's control parameters. From the mathematical model, it can be seen that in (x, y, z, w, m, n) → (x, y, -z, w, m, n), (x, y, z, w, m, n) → (-x, -y, z, -w, -m, -n), and... The transformation (x,y,z,w,m,n)→(-x,-y,-z,-w,-m,-n) results in equation (3) which is invariant and can provide conditions for the formation of symmetric coexisting attractors.

2. The application method according to claim 1, characterized in that, The application method involves adding a constant controller g to equation (4) to obtain a new system model, namely equation (10), as follows: With parameters a = 75, b = 70, c = 2, d = 15, e = 10, f = 1, and initial states X1 = (1, 1, 1, 1, 1, 1) and X3 = (-1, -1, 1, -1, -1, -1), a four-winged attractor is successfully generated when g = -0.8, and the two attractors are symmetrical about the z-axis.

3. The application method according to claim 1, characterized in that, The process of generating the chaotic sequence in step one is as follows: With parameters a = 75, b = 70, c = 2, d = 15, e = 11, f = 1, α = 1, β = 0.01, and time step T = 0.0005, the six-dimensional memristor hyperchaotic system is continuously iterated to generate a hyperchaotic sequence (x(i), y(i), z(i), w(i), m(i), n(i), p, q); The process of performing the scrambling operation in step three is as follows: Here, sum(rows) represents summing the vector formed by row indices, sum(cols) represents summing the vector formed by column indices, floor is the floor operation, m and n are the number of rows and columns of matrix P, respectively, rows and cols are the row vector and column vector of matrix S1, respectively, and S1(i,j) represents the element in the i-th row and j-th column of matrix S1. Through these two formulas, each element P(i,j) in the original matrix P can be mapped to a new position (u1,v1) and a permutation operation can be performed. P r P g and P b The global scramble is performed into three sequences: The process of calculating the output matrix C in step four is as follows: Calculate matrix T for each row i for: Where (i,:) represents all elements in the i-th row of the matrix; Calculate the output matrix C for each column. i for: Where (:,i) represents all elements in the i-th column of the matrix, and F is the pixel value of the grayscale image, F = 256. This indicates a round-down operation.

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