# Synchronization and Application of a Novel Hyperchaotic System Based on Adaptive Observers
## 2 Materials and Methods
### Dynamic Analysis of the Proposed Hyperchaotic System  
The bifurcation diagram of the proposed new hyperchaotic system is shown in Figure 2. While plotting the bifurcation diagram, the parameter \(b\) is taken as 45 and the variation in the parameter \(a\) between 14 and 16 is plotted. For the bifurcation diagram analysis, the initial values of the system are taken as \(x_{1}(0)=1,\ x_{2}(0)=1,\ x_{3}(0)=1,\ x_{4}(0)=1\). As can be seen from the Figure 2, the value of the parameter \(a\) is taken as 15 for the proposed system. Additionally, the 3D bifurcation diagram of the proposed hyperchaotic system is shown in Figure 3. While drawing this bifurcation diagram, in addition to parameter \(a\), the bifurcation diagram of the proposed hyperchaotic system is shown in Figure 3.  
Figure 1: Illustration of the proposed methods. Figure 2: Bifurcation diagram based on \(a\) parameter. parameter \(b\) was also assumed to be between 40 and 50. For the bifurcation diagram analysis, the initial values of the system are taken as \(x_{1}(0)=1,\ x_{2}(0)=1,\ \ x_{3}(0)=1,\ \ x_{4}(0)=1\). Based on Figure 3, the \(b\) parameter value of the proposed hyperchaotic system is taken as 45. The proposed hyperchaotic system can be represented in vector notation as in (2):  
\[\dot{x}=f(x)= \begin{bmatrix}f_{1}(x)\\ f_{2}(x)\\ f_{3}(x)\\ f_{4}(x)\end{bmatrix}  \tag{2}\]  
Based on this, the proposed system can be expressed as in (3):  
\[ \begin{array}{l}f_{1}(x)=-12x_{1}+30x_{2}\\ f_{2}(x)=bx_{1}-x_{2}-x_{1}x_{3}\\ f_{3}(x)=ax_{2}+x_{1}x_{2}-x_{3}+x_{4}\\ f_{4}(x)=-x_{1}b+x_{1}x_{3}-2x_{4}\end{array}  \tag{3}\]  
The divergence of the hyperchaotic system is investigated as (4) for the system [30]:  
\[div\ f=\frac{\partial f_{1}(x)}{\partial x_{1}}+\frac{\partial f_{2}(x)}{ \partial x_{2}}+\frac{\partial f_{3}(x)}{\partial x_{3}}+\frac{\partial f_{4 }(x)}{\partial x_{4}}=-12-1-1-2=-16 \tag{4}\]  
For the divergence of the system, \(div\ f<0\) is required. As can be seen from (4), the system satisfies the condition. To find the equilibrium points of the proposed hyperchaotic system, the system of equations must be equal to zero. This equality is shown in (5):  
\[ \begin{array}{l}0=-12x_{1}+30x_{2}\\ 0=bx_{1}-x_{2}-x_{1}x_{3}\\ 0=ax_{2}+x_{1}x_{2}-x_{3}+x_{4}\\ 0=-x_{1}b+x_{1}x_{3}-2x_{4}\end{array}  \tag{5}\]  
When analyzing the bifurcation diagrams of the system, specific values of \(a=15\) and \(b=45\) were chosen to induce hyperchaotic behavior. Under these parameter settings, the system exhibits three equilibrium points, namely \(E_{1}=[0,0,0,0,0]\),  
Figure 3: The 3D Bifurcation diagram based on \(a\) and \(b\) parameters. \([5.55,2.22,44.6,-1.11]\) and \(E_{3}=[-20.05,-8.02,44.6,4.01]\). To ascertain the stability of the hyperchaotic system, it is imperative to determine the system's eigenvalues. For the specified values \(a=15\) and \(b=45\), the system's Jacobian matrix, essential for calculating the eigenvalues, is presented in (6). \[J= \begin{bmatrix}-12&30&0&0\\ b-x_{3}&-1&-x_{1}&0\\ x_{2}&x_{1}+a&-1&1\\ x_{3}-b&0&x_{1}&-2\end{bmatrix}  \tag{6}\]  
The Jacobian matrix was solved, and the eigenvalues corresponding to the equilibrium points \(E_{1}\), \(E_{2}\) and \(E_{3}\) were determined. Equation (7) was employed for the computation of these eigenvalues:  
\[|J-\lambda I|=0 \tag{7}\]  
Table 1 provides the values of the equilibrium points and eigenvalues. Upon analyzing the eigenvalues (\(\lambda\)) of the system, it becomes evident that the system possesses positive eigenvalues, indicating its instability. Furthermore, it can be inferred that the hyperchaotic attractor of the new system is affiliated with the self-exciting family. The [PERSON] algorithm was employed for the computation of Lyapunov exponents [31]. For the first computation, throughout the calculation, the initial conditions were set as \(x_{1}(0)=1,\ x_{2}(0)=1,\ x_{3}(0)=1,\ x_{4}(0)=1\), parameter \(b\) was taken as 45