Discrete hyperchaotic map is a unique dynamic system, and the key point is to improve the complexity and security of the system with the help of high-dimensional chaos characteristics. Based on the Cubic map, a new discrete hyperchaotic map model can be constructed. Memristors, as the fourth fundamental circuit element, relate charge to magnetic flux and possess unique memory characteristics. In this paper, a cosine-type discrete memristor is proposed, in which the relationship between current and voltage and the relationship between internal charge variables is described as Equation 1: V n = M q n ⋅ I n q n + 1 = q n + I n (1) where M ( q ( n ) ) = c o s ( q ( n ) ) is the periodically varying discrete memory resistance, q is the internal charge variable of the memristor, V and I are the voltage and current of the memristor respectively, and an improved class of discrete memristors can be obtained by increasing the parameters g and constant k of the coupling strength of the cosine discrete memristor, the memristor model is shown in Equation 2: V n = M q n ⋅ I n = k + g ⋅ c o s q n ⋅ I n q n + 1 = q n + I n (2)

Adding the power supply I ( n ) = A s i n ( ω n ) ( ω is the radian frequency) as input to the discrete memristor produces a characteristic volt-ampere graph as shown in Figure 1. The fixed parameters A = 0.1 , g = 1 , k = 1 and q 0 = 0.1 , it can be seen from the figure that the volt-ampere characteristic curve of the discrete memristor is a diasteretic loop diagram in the shape of “8” through the origin point, when A = 0.1 and ω = 0.3 , 0.4, 0.7 are taken, Figure 1a is the volt-ampere characteristic curve of the frequency-dependent tight hysteresis loop shape, from which it can be seen that with the increase of radian frequency, the area of the hysteresis loop gradually decreases, and finally tends to a straight line. When ω = 0.7 and A = 0.1 , 0.15, 0.2, the characteristic curve of the volt-ampere of the discrete memristor is shown in Figure 1b. As the amplitude A decreases, the area of the tight hysteresis loop gradually decreases and finally tends to a straight line. Therefore, its volt-ampere characteristics fully meet the requirements of generalized memristor characteristics.

Through the analysis of numerical simulation, the trajectory distribution and dynamic behavior characteristics of the model can be clearly observed under different initial conditions. It can be seen that the discrete hyperchaotic map not only opens up a new perspective for the research of chaos theory, but also lays a solid theoretical foundation for practical application in related fields.

Compared with traditional chaotic maps, discrete hyperchaotic maps exhibit richer dynamic characteristics in parameter space, including irregular periodicity and extreme sensitivity to initial conditions [46]. In the hyperchaotic mapping system, the memristor interacts with other maps. The nonlinear characteristics of the memristor will be coupled with the characteristics of other components, thus generating more complex nonlinear dynamic behaviors, so as to improve the complexity and robustness of their chaotic behavior.

The cubic map is a discrete chaotic map with a simple structure. Its iterative equation is shown in Equation 3: x n + 1 = a x n 3 − b x n (3)

By introducing the memristor model (Equation 2) into the Cubic map, a new two-dimensional discrete memristive chaotic system can be obtained: x n + 1 = a x n 3 − b 1 + g c o s y n x n y n + 1 = y n + x n (4) When the parameters are a = 0.30 , g = 0.6 , b = 1.50 , the two indices are L E 1 = 0.506156 and L E 2 = 0.0820328 , respectively, the system (Equation 4) has two positive Lyapunov exponents, and the system is in a hyperchaotic state at this time. The phase diagram of its hyperchaotic attractor is shown in Figure 2. As can be seen in Figure 2, the structure of the map is simple, but the dynamic behavior is complex.

where a , b , and g are the control parameters, and in practical applications, the discrete memristor realizes the real-time adjustment of the dynamic behavior in the hyperchaotic map model through its variable resistance characteristics. This mechanism not only improves the adaptability of the system, but also expands the application range of the hyperchaotic map in the field of information encryption.

In the study of chaotic systems, an immobile point is one of the important features of a dynamical system, denoting a state that remains unchanged during the evolution of the system. For a new type of discrete hyperchaotic map, it is of great theoretical and practical significance to determine the location and properties of its fixed points. The fixed point of 2D-DMC is the solution to Equation 5. x ∗ = a x ∗ 3 − b 1 + g ⁡ cos y ∗ x ∗ y ∗ = y ∗ + x ∗ (5) From Equation 5, it follows that 2D-DMC has infinite fixed points, which can be expressed as F = ( x ∗ , y ∗ ) = ( 0 , Q ) , where Q is an arbitrary constant. The characteristic equation of the system can be obtained using the Jacobian matrix of fixed points F as shown in Equation 6. P λ = λ − 1 λ + b 1 + g ⁡ cos ⁡ Q (6)

It can be seen that the eigenvalue λ 1 = 1 always lies in the unit circle, and whether λ 2 lies inside or outside the unit circle depends on the parameters b , g and the internal initial condition Q of the memristor. Therefore, by adjusting the parameters b , g and Q , the fixed point of the 2D-DMC can be placed in an unstable or critical stable state.

The properties of these fixed points determine the complexity of hyperchaotic maps and their potential applications in image encryption. Further studies show that appropriate initial values and parameters can make fixed points exhibit rich dynamic behaviors, thus enhancing the security of chaotic systems.

In general, the study of fixed points provides an important theoretical basis for the application of new discrete hyperchaotic maps. Through in-depth analysis of the properties of fixed points, we can understand the behavior characteristics of chaotic systems and provide valuable guidance for the design and implementation of image encryption algorithms.

In the dynamic analysis, the chaotic characteristics of the model can be evaluated by using tools such as the Lyapunov exponent and the bifurcation diagram. In order to explore the sensitivity of the system to different parameters, the dynamic behavior of the system is analyzed in detail by a bifurcation diagram and the Lyapunov exponential spectrum.

(1) The influence of the parameter b on the system: set the parameter a = 0.3 , g = 0.6 to explore the influence of the system parameter b on the discrete memristor system. The initial state is x 1 = 0.1 and y 1 = 0.1 . In the range of b ∈ [1.1, 1.5], L E s and their bifurcation plots of the discrete memristor chaos map are shown in Figures 3a, b.

As can be seen in the figure, when the parameter b ∈ [1.43, 1.54] range, the system has two positive Lyapunov exponents, indicating that the system is in a hyperchaotic state in this range. For example, when b = 1.48, the phase diagram of the hyperchaotic attractor is shown in Figure 4a. In the range of parameters b ∈ (1.25,1.27), b ∈ [1.28,1.29) and b ∈ [1.3,1.41), the system has a positive Lyapunov index, indicating that the system is in a chaotic state in this range. For example, when b = 1.26 , 1.29, and 1.35, the chaotic attractor of the system is shown in Figures 4b–d. When the parameters are in the range of b ∈ [1.1, 1.25], the system is in a periodic state. For example, when b = 1.15 and b = 1.20 , the chaotic attractor of the system is shown in Figures 4e, f. Through numerical simulation of the model, the rich trajectory behavior and dynamic behavior of the discrete memristor system can be observed under different initial conditions.

(2) Impact of the parameter a on the system: Similarly, to explore the impact of parameter a on the system, parameter b is set to 1.5, g to 0.6, and parameter a varies within the range [2, 5]. The L E s of the discrete memristor chaotic map and its bifurcation diagram are shown in Figure 5.

As can be seen in Figure 5, when a ∈ [2, 3.95) and a ∈ (4.08, 5), the discrete memristor chaotic system exhibits hyperchaotic behavior. For example, when a = 2.5 and 4.5, it can be seen that the system has two positive Lyapunov exponents, and the chaotic attractors of the system are shown in Figures 6a, b. When a ∈ [3.95, 3.98], the system has a periodic attractor. For example, when a = 3.96 , the chaotic attractor of the system is shown in Figure 6c. When a ∈ [4.06, 4.08], the system has a positive Lyapunov exponent and is in a chaotic state. For example, when a = 4.07 , the chaotic attractor of the system is shown in Figure 6d. The analysis of the comprehensive parameter bifurcation diagram and the Lyapunov exponent can provide a solid theoretical basis for the application of new discrete hyperchaotic maps.

(3) The influence of the parameter g on the system: In addition, to explore the influence of parameter g on the discrete memristor system, parameters a = 2.5 and b = 1.5 are set to make the parameter g change in the range of [ 0,0.61 ] , and the L E s and their bifurcation diagrams of the chaotic map of the discrete memristor are shown in Figure 7.

When parameters a = 2.5 , b = 1.5 and initial values ( x 1 , y 1 ) = ( 0.1 , 0.1 ) are selected, the bifurcation plot and LE exponential spectra for parameter g are shown in Figures 7a, b. As can be seen in Figure 7, with the change of parameter g , the discrete memristive chaotic system enters the chaotic state from the typical periodic bifurcation, and a complex window period appears in the chaotic region. When g ∈ [0.31, 0.32], the system has a positive LE exponent and presents a chaotic state, and at g ∈ [0, 0.31) and g ∈ (0.32, 0.32], the discrete memristic chaotic system behaves periodically. For example, when g = 0.42 , the periodic attractor of the system is shown in Figure 7c. When g ∈ [0.57, 0.6], there are two positive LE exponents, and the discrete memristic chaotic system exhibits hyperchaotic behavior. For example, when g = 0.6 , the discrete memristive chaotic attractor of the system is shown in Figure 7d.

The discrete memristive chaotic attractors corresponding to the different parameter values g are shown in Figure 8. It can be observed that the chaotic attractor has a complex fractal structure and with increasing parameter g , the originally separated chaotic attractor, as shown in Figure 8a, gradually merges with the adjustment of system parameters to form a more complex and unique global attractor, as shown in Figure 8d. The synthesis process of chaotic attractors increases the dimension and complexity of the state space of the system, so that the discrete memristive chaotic system can be flexibly applied to the field of information security. In addition, in the discrete memristive chaotic system, the chaotic sequence generated by the composite attractor has better randomness and non-repeatability, and this complex dynamic behavior makes the output sequence of the system difficult to predict, which provides a high degree of nonlinear characteristics for the encryption process and increases the difficulty of cracking.

Discrete memristive hyperchaoticmap has been widely used to improve the credibility of data analysis, random number generation, and encrypted communication. In these areas, randomness is a critical requirement, as the resulting chaotic sequences that do not have sufficient randomness can easily be cracked or predicted, compromising the security of the application. Through the randomness test, the randomness and safety of chaotic sequences generated by the discrete memristive hyperchaotic map can be evaluated. To test the randomness of chaotic sequences, we performed two statistical tests, NIST and 0-1. NIST tests are a series of standardized tests that are used to evaluate and verify the security of random number generators and cryptographic algorithms to check whether the generated data are random. The test results are shown in Table 1, from which it can be found that all P values are greater than 0.01, indicating that the key system has successfully passed the test and the generated data have sufficient security and randomness.

The “0-1 test” generally refers to a statistical test performed on a random number or chaotic sequence, mainly to evaluate whether the resulting sequence is sufficiently random to meet specific statistical requirements and application needs. Firstly, a chaotic sequence with a duration of more than 2000 was randomly selected, and the values were selected at a certain step interval for testing, and the test results are shown in Table 2. As you can see from the results in the table, the test result value is close to 1. This indicates that the discrete memristive hyperchaotic map exhibits a high degree of randomness.

Based on the results shown in Tables 1, 2, it can be concluded that the key system derived from the discrete memristive hyperchaos map has excellent randomness. Chaotic randomness testing is of great significance in application security, data analysis, and simulation, which can ensure the security of the application and meet the security and reliability requirements required for image encryption.

In this section, a novel image encryption scheme based on a two-dimensional hyperchaotic map based on cyclic shift, forward and reverse diffusion, and global displacement is introduced. By combining key steps such as pixel diffusion and displacement, hyperchaotic sequences are used to reorder the pixel positions of the original image and disrupt the overall structure of the image. On the one hand, the displacement process ensures randomness, while diffusion further enhances the complexity of image encryption. The processed chaotic sequence and pixel value are used to perform XOR operations to further improve the encryption strength and reduce the risk of information leakage. The steps are as follows:

1. Select the original image and perform channel separation, and select a grayscale image of m × n as the original image. m × n chaotic sequences X ( m ) , Y ( n ) are generated from the state variables x 0 , y 0 , and two chaotic sequences X ( m ) and Y ( n ) are generated using a Gaussian chaotic neural network, which is used for row and column shifts, respectively.

2. Generate m × n chaotic sequences from the state variables x 0 , y 0 The chaotic sequences Z ( m , n ) and the scrambled image are added pixel by pixel to achieve positive diffusion. Regenerate m × n chaotic sequences W ( m , n ) from state variables x 0 , y 0 . The chaotic sequence W ( m , n ) and the image after forward diffusion are subtracted pixel by pixel to achieve reverse diffusion.

The image encryption algorithm of the new discrete memristive chaotic system provides a secure and powerful encryption scheme for grayscale image encryption. In the encryption stage, the plaintext image undergoes cyclic shift, forward and reverse diffusion, and global substitution operations, combined with the dynamic key generated by the chaotic system, and finally generates an irreversible ciphertext image,as shown in Figure 9a. During decryption, global substitution, forward and reverse diffusion, and cyclic shift are performed in reverse, and the original pixel value and position are restored by the same chaos key to achieve lossless decryption,as shown in Figure 9b.

In image encryption algorithms, the size of the key space directly determines the security of the encryption. To verify the feasibility and effectiveness of the proposed algorithm, simulation tests were performed using Matlab 2023b, with the key set as g = 0.6 , a = 3 , b = 1.8 and ( x 0 , y 0 ) = ( 0.1 , 0.1 ) . A chaotic sequence required for encryption was generated using a discrete memristive chaotic system, and then the image was encrypted through the encryption algorithm. When establishing a new discrete hyperchaotic map, the selection of the key depends on multiple parameters, such as initial conditions and the dynamic characteristics of the system. As shown in Figures 10a, d, g is the original image before the above-mentioned algorithm is encrypted, Figures 10b, e, h are the encrypted image after using the above-mentioned encryption algorithm, and Figures 10c, f, i are the decryption image after using the above-mentioned algorithm. In order to verify the security of the encryption effect of the system, this paper conducted performance analysis, mainly including key space analysis, histogram analysis, correlation analysis, and information entropy analysis.