## 2 2D Hyperchaotic Map  
An electric field-controlled memristor uses an electric field to regulate its resistance value. Typically, it consists of a dielectric layer and two electrodes. When the size of the electric field changes, the degree of polarization in the dielectric also changes, which causes the resistance value of the memristor to change.  
\[I=I_{0}\times\exp(-\frac{g}{g_{0}})\times\sinh(\frac{V}{V_{0}}) \tag{1}\]  
where \(V\) is the voltage, \(I\) is the current, and \(g\) denotes the gap between the upper and lower electrodes.  
In a compact model, the oxide layer is positioned between the two electrodes, and the resistance strongly correlates with the gap distance (denoted as \(g\)) between them. The average distance \(g\) is used to determine the resistance of the RRAM through the electron tunneling conduction mechanism, resulting in an exponential increase in resistance with gap distance [22]. When a voltage (\(V\)) is applied to the memristor, a non-linear relationship between the resistance value and the applied voltage emerges. The resistance value decreases, remains stable, or increases with the voltage, and the current changes accordingly. Consequently, the current-voltage (\(I\)-\(V\)) relationships of the RRAM model are expressed as Equation (1). As nonlinearity is one of the key factors that cause chaos, it is considered to incorporate the memristor model into the chaotic system.  
The inherent nonlinearity, complexity, and randomness of this memristor model used in the chaotic model building usually help to generate chaotic behavior, which can improve security and randomness in chaotic image encryption, thus providing a better guarantee for the confidentiality of image data [23; 24]. Building upon this premise, this paper modifies the model and integrates it with trigonometric functions to introduce a novel 2D hyperchaotic map. This mapping can be expressed mathematically as Equation (2).  
\[\begin{cases}x(i+1)=cos(asin(\tau x(i)/4)+3\times exp(-2x(i))sinh(by(i)))\\ y(i+1)=cos(asin(\tau y(i)/4)+3\times exp(-2y(i))sinh(bx(i)))\end{cases} \tag{2}\]  
Here, \(a\) and \(b\) represent system parameters, with \(b\) greater than 3.5 or less than \(-3.5\), and \(x(0)\) is not equal to \(y(0)\).  
Through further discussion, the universal structure of this chaotic system is established as follows in Figure 1, where the function F represents the trigonometric function, the function G and the function H represent the nonlinear memristor model, such as the chaotic system function G mentioned above represents the \(3\times exp(-2x(i))sinh(by(i))\).  
\[\begin{cases}x(i+1)=sin(acc(x(i))+3\times exp(-2x(i))sinh(by(i)))\\ y(i+1)=sin(acc(y(i))+3\times exp(-2y(i))sinh(bx(i)))\end{cases} \tag{3}\]  
In order to further verify the validity of the proposed structure, a map can be mathematically represented by Equation (3). The phase diagram, bifurcation diagram, and  
Figure 1: System framework diagram.  
Lyapunov exponential spectrum of the system are briefly analyzed as shown in Figure 2, where \(\lambda 1\) and \(\lambda 2\) represent the two Lyapunov exponentials, and data1 represents the 0. It is shown that the system also has the optional large parameter range and complex chaotic behavior.