# A Two-Dimensional mKdV Linear Map and Its Application in Digital Image Cryptography
## 2 Formulation of the Problem
### Construction of a Two-Dimensional Linear Map (Case Study: Sine-Gordon Map)
Figure 4: Cipher images after using Arnold’s Cat Map (ACM) (**a**) and sine-Gordon (**b**) map obtained from encrypting the original image in Figure 2 with Equation (24) and Equation (23). _Bifurcations_. The word \"bifurcation\" refers to a sudden qualitative change in the nature of solutions that occurs as the parameter is varied. The bifurcation diagram shows many sudden qualitative changes in the attractor as well as in the periodic orbits [29]. Considering the level set diagrams in Figures 5-7, when \(\alpha\) varies, the critical points of \(H(x,y:\alpha)\) change from an elliptic type to a saddle type. A known mechanism in the literature, for integrable systems, is through a Saddle-Center bifurcation, where one saddle point becomes degenerate, and breaks into three critical points: two saddles and one elliptic (or also known as center) point [26]. In the case of \(H(x,y:\alpha)\), the bifurcation diagram with respect to varying of \(\alpha\) is shown in Figure 8. Note that the blue dots that spread out represent the chaotic region. _The Maximum Lyapunov Exponent_: The chaotic behavior of a dynamical system is defined by its sensitiveness to initial conditions and unpredictability. The Lyapunov exponent (LE) is a quantitative measurement to determine the dynamical system's chaotic behavior [30]. LE measures the degree of divergence between any two nearby trajectories of a dynamical system, stretching an orbit. The positive value of maximum LE shows exponential stretching of an orbit. Every chaotic system has at least one positive Lyapunov exponent to achieve chaotic behavior. A multidimensional chaotic map with more than one LE, and if all the LEs are positive, then the dynamical systems possess hyperchaotic behavior. The hyperchaotic behavior  
Figure 8: A bifurcation diagram of \(H(x,y:\alpha)\) for \(x\in[-3,3]\) and \(\alpha\in[-10,10]\). Figure 7: The level set of \(H(x,y:\alpha)\) for \(\alpha=0.1249\). The fixed point \((x_{6},x_{6})=(1.34332,1.34332)\) is unstable. is good for image encryption and it is very difficult for the attacker to predict the secret key [31]. For two-dimensional map \((x^{\prime},y^{\prime})=(f_{1}(x,y),f_{2}(x,y))\), in Equation (12) for example, we can calculate the maximum LE, \(\lambda^{\max}(x_{0},y_{0})\) at the point \((x_{0},y_{0})\) by using the following formula [29]:  
\[\lambda^{\max}(x_{0},y_{0})\approx\log\sqrt{\left(\frac{\eta(x_{0},y_{0})+ \sqrt{4(\kappa(x_{0},y_{0}))^{2}+(\vartheta(x_{0},y_{0}))^{2}}}{2}\right)} \tag{25}\]  
where  
\(\eta=(\partial_{x}f_{1}(x_{0},y_{0}))^{2}+\big{(}\partial_{y}f_{1}(x_{0},y_{0}) \big{)}^{2}+\big{(}\partial_{x}f_{2}(x_{0},y_{0})\big{)}^{2}+\big{(}\partial_{ y}f_{2}(x_{0},y_{0})\big{)}^{2}\),  
\(\kappa=(\partial_{x}f_{1}(x_{0},y_{0}))^{2}-\big{(}\partial_{y}f_{1}(x_{0},y_{ 0})\big{)}^{2}+\big{(}\partial_{x}f_{2}(x_{0},y_{0})\big{)}^{2}-\big{(}\partial _{y}f_{2}(x_{0},y_{0})\big{)}^{2}\),  
\(\vartheta=(\partial_{x}f_{1}(x_{0},y_{0}))\big{(}\partial_{y}f_{1}(x_{0},y_{ 0})\big{)}+\big{(}\partial_{x}f_{2}(x_{0},y_{0})\big{)}\big{(}\partial_{y}f_{ 2}(x_{0},y_{0})\big{)}\). Now, we can use \(\lambda^{\max}(x_{0},y_{0})\) to acquire the maximum LE for 2D nonlinear mKdV map in Equation (12). For \(\alpha=-9.79\) and the fixed point \((x*,y*)=(2.5571,2.5571)\), we have  
\[\eta=1+\left(-\frac{(-9.79-x)}{(-1+x)x^{2}y}-\frac{(-9.79-x)}{(-1+x)^{2}xy}- \frac{1}{(-1+x)xy}\right)^{2}+\frac{(-9.79-x)^{2}}{(-1+x)^{2}x^{2}y^{4}},\]  
\[\kappa=1+\left(-\frac{(-9.79-x)}{(-1+x)x^{2}y}-\frac{(-9.79-x)}{(-1+x)^{2}xy}- \frac{1}{(-1+x)xy}\right)^{2}-\frac{(-9.79-x)^{2}}{(-1+x)^{2}x^{2}y^{4}},\]  
\[\vartheta=-\left(\frac{(-9.79-x)\bigg{(}-\frac{(-9.79-x)}{(-1+x)x^{2}y}-\frac {(-9.79-x)}{(-1+x)^{2}xy}-\frac{1}{(-1+x)xy}\bigg{)}}{(-1+x)xy^{2}}\right)\]  
Substituting \((x,y)=(2.5571,2.5571)\), we get \(\eta=2.55876,\kappa=2.1089\), and \(\vartheta=0.547745\). Finally, we have the maximum LE number is \(\lambda^{\max}(x_{0}=x*)\approx 0.451625\). Thus, the orbit of the mapping in Equation (12) is exponential stretching.